Simmons and Su's equivariant Tucker-type conjecture for $S_k^{n(k-1)}$

Let

$$S_k^{n(k-1)}= \left\{(z_0,\ldots,z_{n(k-1)})\in\mathbb C^{n(k-1)+1}:\sum_{i=0}^{n(k-1)}|z_i|=1,\ z_i^k=|z_i|^k\right\},$$

where $\omega=e^{2\pi i/k}$, and let the symmetric group $\operatorname{Sym}(k)$ act on this space by permuting the $k$th roots of unity in each coordinate. Suppose the space is triangulated invariantly under this action, with vertex set $V$ labeled equivariantly by

$$\ell:V\longrightarrow\{\omega^j m:1\leq j\leq k,;1\leq m\leq n\},$$

meaning that $\ell(\pi(v))=\pi(\ell(v))$ for every $\pi\in\operatorname{Sym}(k)$ and $v\in V$.

Simmons and Su's conjecture. There must exist $k$ adjacent vertices in the triangulation whose labels are $\{\omega^j m:1\leq j\leq k\}$ for some fixed $m$.

This is an equivariant, symmetric-group version of Tucker's lemma and is intended as a combinatorial counterpart to the Borsuk-Ulam property. The supplied text presents it as a conjecture of Simmons and Su but gives no resolution, so its status is recorded as open.