Lovász's conjecture on the Stiefel–Whitney height of odd-cycle Hom complexes

Let $C_{2r+1}$ be the cycle graph of odd length $2r+1$, with $r\geq 1$, and let $K_n$ be the complete graph on $n$ vertices. For a graph $T$, write $\operatorname{Hom}(T,K_n)$ for its Hom complex, equipped with the involution induced by the relevant $\mathbb Z_2$-action, and let $\varpi_1(\operatorname{Hom}(T,K_n))$ denote its first Stiefel–Whitney class. The equation under consideration is

$$\varpi_1^{n-\chi(T)+1}(\operatorname{Hom}(T,K_n))=0,\qquad n\geq \chi(T)-1.$$

Lovász's conjecture. The equation is true for $T=C_{2r+1}$, $r\geq 1$, and all $n$.

The claim extends the cases established in the source for complete graphs and for odd $n$. The preceding discussion verifies the case $n=3$ by analyzing the connected components of the Hom complex; the general assertion is the stated conjectural extension.