Hm. I expected as much. Close, but not quite what's needed.
It's pretty much impossible for any add-on to be able to anticipate every possible desired result.
Please try it again, this time with this yeargroup tpl also.
<ul>
<li>
<ul>
[[+row]]
</ul>
</li>
</ul>
What we want to do is to get the outer year, then a list of inner months.
That list of inner months can be filtered with an output filter to only display the ones whose date contains the same year as the outer year item. Or at least I'm pretty sure this would work. It's just a matter of getting the levels of tpls sorted out.