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In 2016, the cabins in that area will open on June 23, according to the Hemavan Tourist Office:
Hi David,
They open a few days later that month, the 23 of june so if you can wait a few more days it is not a bad idea. Before that it can still be a lot of snow and/or very wet in the terrain so we recommend people to walk after midsummer, usually it is better conditions then.
Best regards / Med vΓ€nlig hΓ€lsning
Further north (Abisko to Kebnekaise), the cabins actually open a few days earlier:
The huts [near Abisko] open on June 17th. Temperatures have been really high the last weeks so chances are that the trail will be snow-free even in higher altitudes by the time the huts open. However, I would recommend contacting [email protected] for more information.