A reduction conjecture for the complexes $\Gamma_n^{\alpha,r}$

For fixed $m>0$, let $\mathcal{G}_m^n$, $\Delta_m^{n,r}$, and $\Gamma_n^{\alpha,r}$ be the complexes defined in the paper. Let $V(\Delta)$ denote the vertex set of an induced subcomplex $\Delta$ of $\Gamma_n^{\alpha,r}$. Reduction conjecture. If $r\geq n\geq 2$, then $\Gamma_n^{\alpha,r}$ is homotopy equivalent to an induced subcomplex $\Delta$ such that every $x\in V(\Delta)$ satisfies

$$|x_i|<\left\lfloor\frac{r}{2}\right\rfloor\quad\text{for all }i\in[n]$$

and, for every $S\subseteq[n]$ with $\operatorname{Card}(S)=n-2$,

$$\sum_{i\in S}|x_i|\leq r-2.$$

The conjecture is proposed as a possible generalization of the reduction lemmas used in the paper. Its resolution would provide a systematic homotopy reduction for these complexes in arbitrary dimension.