The winding-number conjecture for dilated simplex Ehrhart h*-vectors

Let $\Delta_r^n=\{x\in[0,r]^n:\sum x_i=r\}$ be the $(n-1)$-dimensional simplex, and call a decorated ordered set partition $((L_1)_{l_1},\ldots,(L_k)_{l_k})$ simplicial when $1\le l_i\le r-1$ and $\sum_i l_i=r$. For a simplicial ordered set partition with $1\in L_1$, let $w_P$ denote its winding number. Denote by $m_0,m_1,\ldots$ the numbers of such partitions having winding numbers $w_P=0,1,2,\ldots$. The dilated-simplex winding-number conjecture. The vector $(m_0,m_1,\ldots)$ equals the Ehrhart $h^*$-vector of $\Delta_r^n$, and is obtained from the generating series

$$\sum_{s=0}^{\infty}\binom{n-1+rs}{n-1}x^s.$$

The conjecture proposes a combinatorial interpretation of the Ehrhart $h^*$-vector, extending the computationally observed winding-number enumeration for dilated simplices.