Conjecture on the independence complex of a categorical product of three complete graphs

Let $I(G)$ denote the independence complex of a graph $G$, let $K_n$, $K_m$, and $K_l$ be complete graphs, and let $f(n,m,l)$ denote the unspecified number of spheres occurring in the proposed homotopy type. Three-factor complete-graph conjecture.

$$I(K_n\times K_m\times K_l)\simeq\bigvee_{f(n,m,l)}\mathbb{S}^3.$$

This conjecture proposes a uniform description for the independence complexes of categorical products of three complete graphs. The source presents it as a conjecture but does not provide, in the supplied text, evidence resolving the general statement.