@PiotrPstrągowski i studied these in my thesis (the kan--quillen model structure on sSpaces). i remember trying to work out this example, and not being able to -- apparently i left this as Speculation I.2.35. but i now think it should be true. here's a plausibility argument. to check this, it suffices to look at simplicial objects in grouplike $A_\infty$-spaces, and these can be presented by simplicial groups. [cont.]

moreover things should be nice enough (combinatorial, whatever) so that we know that simplicial objects in the $\infty$-category are presented by simplicial objects in the model category. then, use the usual argument (that all horn inclusions admit sections after applying the "levelwise free group" functor).