Hi charliethefig,
All humid (leaves and fruits are) material you leave on a protected tree is very likely to rot and you don't want the stems to rot.
It happened to me last year on a tree - I missed that bloody fig - and one fig got against the higher part of the main trunk and that created a brown stain and voilà the tree after that stain dried out and died.
The tree grew from under that zone, but guess I lost some 10 brebas with that forgotten fig.
Normally here, the figs get toast and fall, but as last winter was some mild, some figs made it through the winter on the trees, and that allowed for that fig to stay against the stem.
I left some of those figs as a test to know if they were worth it here in my Zone7, but experience showed me that although they made it through, they had brown stains from frost damage,
so for me, not worth leaving them on the trees.
I've been knocking figs on trees since end of September, but some trees did grow new figs since that date, and so I'm still knocking figs.
I, as well, left on purposes some figs on the trees to slow down the trees , especially on my "Dalmatie tree 2" .
Or she could have gone vegetative growth and that would have been even worst .
I've been knocking some yellow leaves on the trees for them not to rot on the trees, and I'll wait until we get -5°C at night for some days, to clean completely the trees, let the stems dry a bit (I don't want them wet at this point )
and wrap them and put the 80 liters trashcans at their base.
I have some trees that are now all yellow, while some others are still full green :( .
So, yes, before cocooning remove all humid material or be prepared for mold - and remember humid/wet + mild/heat = mold.