Amount of snow/ice in the Alps
This is not easy to answer. In the Brauneck area you will not expect any snow end of June. If you approach the Karwendel (after six days) I am pretty sure you will have snow when climbing the Birkkarspitze (2.749m) beginning of July. The amount of snow left depends on the weather the next four weeks - nobody knows. It is always recommended to call the Karwendelhaus asking for the condition of the trail through the Schlauchkarsattel. But I do not know if they speak English.
Fave fun