at the moment the TooltipTemplateContext isn't very useful and hard to work with:
Problem: As you can see, "start" and "end" only contains a weired pre-build string.
As a developer I can't parse this (at least the year is missing), so I'd like to suggest, that you replace the Dictionary of <string,string> with an <string,object> and add the start and end as DateTime.
An alternative would be to pass a custom parameter or have the task item accessable directly.
In our use-case we have no need for the time (it's a gantt not a time table!) and the tasks won't end at the end of the year.