Link collapsibility conjecture for independence complexes of bounded-degree graphs

Let $G=(V,E)$ be a graph with maximum degree at most $\Delta$, and let $I_n(G)$ be its independence complex. For an independent set $A$ of size $n-1$, let $\operatorname{lk}(I_n(G),A)$ denote the link of $A$ in $I_n(G)$. Let $C(X)$ denote the collapsibility number of a simplicial complex $X$, and let $n\geq 1$.

Link collapsibility conjecture. If $A$ is an independent set of size $n-1$ in $G$, then

$$C(\operatorname{lk}(I_n(G),A))\leq\left\lfloor\frac{(n-1)\Delta}{2}\right\rfloor.$$

This is presented as a weaker result that may hold for general bounded-degree graphs, after the stronger global collapsibility bound was disproved. Its status is open in the supplied text.