The winding-number conjecture for generic cube cross-section h*-vectors

Fix integers $r,n$ and $s$ with $1\le s\le rn-1$, and let

$$I_{r,s}^n=\left\{x\in[0,r]^n:\sum_{i=1}^n x_i=s\right\}.$$

Consider decorated ordered set partitions $((L_1)_{l_1},\ldots,(L_k)_{l_k})$ with $1\in L_1$, satisfying

$$l_1+\cdots+l_k=s,\qquad 1\le l_i\le r|L_i|-1.$$

Let $w_P$ be the winding number, and denote by $m_0,m_1,\ldots$ the numbers of these partitions having winding numbers $w_P=0,1,2,\ldots$. The cube-cross-section winding-number conjecture. The vector $(m_0,m_1,\ldots)$ equals the Ehrhart $h^*$-vector of the hyperplane cross-section $I_{r,s}^n$. This would provide a combinatorial interpretation of the $h^*$-vectors of integer-sum hyperplane cross-sections of cubes; the source presents it as an additional conjectural extension of the hypersimplex and dilated-simplex cases.