The independence-complex conjecture for products of three complete graphs

Let $K_n$ denote the complete graph on $n$ vertices, let $\times$ denote the graph product, and let ${\rm{Ind}}(G)$ denote the independence complex of a graph $G$. For $n\geq 2$, the independence-complex conjecture.

$${\rm{Ind}}(K_2\times K_3\times K_n)\simeq\bigvee\limits_{(n-1)(3n-2)}\mathbb{S}^3.$$

This conjecture is based on computer calculations of the Betti numbers for the indicated independence complexes; its resolution would establish the proposed wedge-of-spheres description for this family.