Let

$sd : sSet \to sSet$ denote barycentric subdivsion;

$cosk_1 : sSet \to sSet$ denote 1-coskeletalization.

**Question:** Let $X$ be a graph or simplicial set. If the homotopy type of $cosk_1(X)$ is known, then what can be said about the homotopy type of $cosk_1(sd(cosk_1(X)))$?

I'm interested in this question primarily in the case where my simplicial sets are in fact abstract simplicial complexes. This special case of the question can be read as follows. Let

$sd X$ denote the barycentric subdivision of an abstract simplicial complex $X$;

$cosk_1(X)$, for an abstract simplicial complex $X$, denote the largest abstract simplicial complex with the same vertices and edges as $X$.

**Question (bis):** Let $X$ be a graph or abstract simplicial complex. If the homotopy type of $cosk_1(X)$ is known, then what can be said about the homotopy type of $cosk_1(sd(cosk_1(X)))$?