Consider pronilpotent completion $\widehat F$ of a free group $F(X)$: $\varprojlim_k F(X)/F(X)_k$. Is it true that it is projective (naive definition makes sense because epis in $Grp$ are regular, hence universal) object in category of pronilpotent groups considered as full subcategory of $Grp$?
It seems to me that the answer is «yes if $X$ finite, no otherwise», but I'm stuck with a problem of carrying epis through limit for finite case and trying to adjust proof of non-freeness of $\Bbb Z ^{\Bbb Z} $ for inifinite one.