Do we have conventional knowledge of the present?

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The quote you give:

The cognition of the ultimate nature of things—their all being empty of intrinsic nature—is nonconceptual because, there being nothing to cognize, no cognition arises.

Refers to wisdom directly realizing emptiness. Such a wisdom realizes emptiness in a non-dualistic manner. It is non-conceptual and explicit, since it realizes its object (emptiness) by way of emptiness appearing to it. Conceptual consciousnesses realize their object by way of a mental image appearing to them. Cognition arise (this wisdom is a cognition) but there is no acertainment of conventional truths (such as time, a table, persons, the basis of emptiness, and so forth).

By the way, Garfield is not the safest source.

Chandrakirti (same view as Buddhapalita) also talked about "non-seeing" and "non-cognition" or "non-perception." He does not mean that a buddha is free from cognition (that would contradict the typical Mahayana claim hat he is omniscient) but simply that non-existent do not appear to a buddha. A buddha is free from the appearance of true existence. It is not so much about "no cognition" as it is about "no cognition of something in particular."

It is explained in Chandrakirti's Commentary on the "Supplement to Nagarjuna's Treatise on the Middle'.

Geshe Jamphel Gyaltsen said:

When said that they perceive it in the manner of non-perception, it does not mean that a buddha does not see anything at all. It means that he is not seeing any phenomenon arisen due to ignorance, that is, together with a false appearance of the conventional - an appearance of true existence. It is the omniscience realizing the final reality that perceives the emptiness, the absence of true existence. Since a buddha does not perceive what does not exist and knows all existing phenomena, when a truly existent conventional phenomenon were to exist, it should appear to a buddha, but it does not. What is not perceived by him is a non-existent. If it were existent it would appear to the Buddha. True establishment does not exist and therefore does not appear to the wisdom of a Buddha realizing the final reality.

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Finally... let me try。。。

What is the use of discussing no-cognition as it defined as "the non-existence of a thing cannot be perceived by the senses for there is nothing with which the senses could come into contact in order to perceive the non-existence." wikipedia/Anupalabdhi

It negates the negated, i.e., the non-existence, the cannot be perceived. This is from Kumārila Bhaṭṭa (700AD) the Hindu philosopher. This rendered the whole sytax just as a don't-know. Now some scholars use it to fit into the Buddhist concept, or to explain the Buddhist concept, it is an error, IMO.

The cognition of the ultimate nature of things—their all being empty of intrinsic nature—is nonconceptual because, there being nothing to cognize, no cognition arises. --Garfield and Westerhoff in Madhyamaka and Yogacara

The Buddha/Nagarjuna doesn't phase Emptiness in this way. If there are things? Ultimate nature? Cognition or to cognize is a conditioned faculty, same as the 5 Aggregates conditioned by the human body and mind as in an Ordinary (凡夫). The quoted phrase is produced by an academic mind conditioned by education with those technical terms in prescribed syntax.

Eckel in To See the Buddha...

Again, object, reality, knowledge, these terms in such way of a paragraph written doesn't relate anything to what the Buddha taught, although it claimed Buddha here and Buddha there in the sentences. It is an un-beneficial hard work to explain how it was not making sense, same as to explain why a 1 or 2 or any number is not a 0.

And obviously the heart sutra...

Your understanding the Heart Sutra is far from getting it. When in the Heart Sutra it said:

No Suffering Samudaya Extinguishing Path, no knowledge also no attainment. {無苦集滅道無智亦無得}

It is not a negation to describe "Ultimate Reality", in your term. It is because "In Emptiness it has no [such and such]... "{是故空中無...}, for these are dharma(s){法}, i.e., the laws created by us. Not the laws in law books, but laws of nature, physics, E=MC", law of gravity... all are dharma(s) - 法[則] - laws, the products of the intellect/mind. Therefore the Buddha said all knowledge of this world are "games theories" {戲論}, i.e., we set the rules, then we play against these rules. The Heart Sutra, of import is the beginning, "Observe the Self-present Bodhisattva, when professing in deep Prajnaparamita..." {觀自在菩薩行深般若波羅蜜多時...}, this is the condition a Dhyana practitioner is able to enter, then it goes on to give more instructions and describe the experiences etc... This is a long explanation and belongs to another topic I believe. Of note is almost all translated the beginning as Avalokiteshvara, it maybe so or it could be not as so. There is another definite name for this Bodhisattva in Chinese yet here this could refer to Him, or refer to anyone as long as this person who is able to reach the Self-present{自在}, one who is free, totally at ease, free from any bondage or restrain, mindful and, present. (see, this is the perfect condition one should enter during meditation.)

To conclude, I thought your formulating this question is not so beneficial in understanding the Buddhist Dharma, Emptiness, or the philosophy you intended. For it is like saying, does the cow have horse face? {牛頭不對馬嘴}. With no offence, if it does irritate you, apologize for my commend of language, since English is not my mother tongue.

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