Collapsibility bound for independence complexes of bounded-degree graphs

Let $G$ be a graph, and let $I_n(G)$ denote its complex of independent sets of size at most $n$. For a simplicial complex $X$, let $C(X)$ denote its collapsibility number. Let $\Delta$ be a nonnegative integer and let $n$ be a positive integer.

Collapsibility conjecture. If $G$ has maximum degree at most $\Delta$, then

$$C(I_n(G))\leq\left\lceil\frac{\Delta+1}{2}\right\rceil(n-1).$$

This is proposed as an extension of the Aharoni–Briggs–Kim–Kim conjecture. The bound is known for claw-free graphs, but the paper gives counterexamples in general, so the conjecture is refuted.