DataGird with no width set. Resize parent window so that only leftmost column is visible. Maximize the window.
Then all other columns being hidden before maximization are still invisible. 

Workaround: Set width of the DataGrid.

Currently, the columns provide a SortDescriptor and GroupDescriptor properties. Add also, a FilterDescriptor property for the column objects.

This should allow you to enable filtering also in the DataGridTemplateColumn.