The Hom-complex realization conjecture for diameter-one graphs

Let $X$ be a finite simplicial complex, let $T$ be a finite connected graph, let $K_2$ be the complete graph on two vertices, and let $G_{1,X}$ be the graph obtained from $X$ by the construction in the paper with $k=1$. For a graph $H$, write $\operatorname{Hom}(K_2,H)$ for its Hom complex.

Hom-complex realization conjecture. If

$$\operatorname{diam}(T)=1,$$

then there is a homotopy equivalence

$$\operatorname{Hom}(K_2,G_{1,X})\simeq X.$$

This is the precise formulation of the proposed improvement from the general construction to the case of a complete graph with possibly some loops. The supplied text gives no resolution status, so the conjecture remains open.