Equality conjecture for the dimension of cone-volume fibers

Let $U \in \mathcal{U}(n,m)$, and let

$$U=S_1\cup S_2 \cup \cdots \cup S_d$$

be its unique partition into irreducible sets. Let $\gamma \in C_{\operatorname{cv}}(U) \cap \mathbb{R}_{>0}^m$, and define

$$S(U,\gamma)=\\{b\in\mathbb{R}_{\geq 0}^m:\gamma(U,b)=\gamma\\}.$$

Here $\dim^*(S(U,\gamma))$ denotes the dimension of this semialgebraic set. Equality conjecture. The dimension satisfies

$$\dim^*(S(U,\gamma))=d-1.$$

The preceding proposition proves the lower bound $\dim^*(S(U,\gamma))\geq d-1$; the conjecture asserts that this bound is always sharp, describing the expected dimension of the set of right-hand sides yielding a fixed cone-volume vector.