Magic positivity conjecture for generalized parking-function polytopes

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, let $\mathfrak{X}_n(\mathbf{b})$ denote the $\mathbf{b}$-parking-function polytope, the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. A sequence $(\beta_1,\dots,\beta_n)$ is a $\mathbf{b}$-parking function if its nondecreasing rearrangement $\beta'_1\le\cdots\le\beta'_n$ satisfies $\beta'_i\le b_1+\cdots+b_i$. A polytope is magic positive when all coefficients of its magic transform are nonnegative. Magic positivity conjecture. For every $n\ge3$ and every $\mathbf{b}\in\mathbb{Z}_{>0}^n$, the polytope $\mathfrak{X}_n(\mathbf{b})$ is magic positive. Equivalently, the classical parking-function polytope $PF_2=\mathfrak{X}_2(1,1)$ is the only $\mathbf{b}$-parking-function polytope that is not magic positive. The theorem in the paper establishes the claim for the two-parameter family $\mathbf{b}=(a,b,\dots,b)$, and computations verify it for several additional parameter vectors; the case of arbitrary $\mathbf{b}$ remains open.