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According to your Wikipedia page:
In the ancient Roman legend of the kingdom era, the Horatii were triplet warriors who lived during the reign of Tullus Hostilius. The accounts of their epic clash with the Curiatii and the murder of their sister by Publius, the sole survivor of the battle, appear in the writings of Livy.
Who is Tullus Hostilius supposed to be?
Myth and history
As with those of all the early kings of Rome, the events ascribed to the reign of Tullus Hostilius are treated with skepticism by modern historians. Part of this is due to obvious flaws in the literary tradition describing the kings: much like the confusion the Ancients exhibited in attributing identical accomplishments to both Tarquinius Priscus and Tarquinius Superbus, the accomplishments of Tullus Hostilius are thought by many scholars to be rhetorical doublets of those of Romulus.
Let me emphasise that again in force, since Wikipedia is a bit too diplomatic there: even the early republic left no reliable written sources for us. The kingdom era is even more so much less reliable than that. Shrouded in myth, legend and fantasy. There may be traces of actual history in it. But that cannot be ascertained. You cannot take one single thing or fact from the monarchical period and treat it at face value.
Adding to that that the picture in question is more than 2000 years younger than the "event" displayed and is typically for that style inventing its inventory in terms of clothing and weapons, it only presents an idea and allegory. Nothing in it can claim any historical accuracy.
But taking aside the fundamental aspect of "there were no real Horatii": this picture displays a scene that is not to be found in the sources we have, legendary as their factual content is. The absence of the scene from Livy means that the whole setup is made up by the painter as a form of interpolation.