7 problems

Geometric polynomial conjecture. For any affine Weyl group, the map $q_a$ is a geometric polynomial.

Puppe and Slinko's conjecture. A MUCD is connected if and only if it is a peak-pit domain.

Disconnectedness conjecture. Asymptotically almost surely, MUCDs are not connected.

Let $P_n$ be the permutohedron, and let $H_0$ be its affine hull. For an affine hyperplane $H\ne H_0$, consider the number of vertices in $H\cap P_n$. Maximum intersection conjectu…

Let $P_n$ be the permutohedron with vertex set contained in the hyperplane … Let $H_1,\ldots,H_m$ be affine hyperplanes different from $H_0$ whose union contains the vertex set of…

Let $\Pi_d$ denote the regular permutohedron, the usual permutohedron associated with $(1,2,\dots,d+1)$. The BV-$\alpha$-valuation is the valuation used in the paper's McMullen-typ…

BV-alpha positivity conjecture. Every regular permutohedron is BV-$\alpha$-positive.