Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One of the necessary consditions is that the tangent space $t_{\mathcal{F}}:=\mathcal{F}(k[\epsilon]/(\epsilon^2))$ is a finite dimensional $k$-vector space, where $k$ is the residue field of $\Lambda$.
A generalization or maybe an analogue of this is the derived Schlessinger criterion, due to mainly Lurie if I'm not mistaken (see https://nms.kcl.ac.uk/ashwin.iyengar/lntsg2019/G8.pdf Thm 3.5). If I'm not mistaken, in this case we allow the tangent space to be possibly infinite, by which I mean that $H_0(\mathfrak{t}\mathcal{F})$ can be infinite dimensional.
So if I start with a "nice" functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$, whose tangent space is not finite dimensional, can the induced functor $$s\mathcal{F}:sC_{\Lambda}\rightarrow s\text{Set}$$ be pro-representable? In a more optimistic sense, if $\mathcal{F}$ does satisfy every condition of Schlessinger's criterion except the finiteness of $t_{\mathcal{F}}$, is $s\mathcal{F}$ pro-representable?
