Early's winding-number formula for the Ehrhart h*-vector of the hypersimplex

A decorated ordered set partition consists of an ordered partition of $[n]$ whose blocks are decorated by positive integers; it is hypersimplicial of type $(k,n)$ when each block $L_i$ with decoration $l_i$ satisfies $1\leq l_i\leq |L_i|-1$. The winding number is the integer $d$ determined by the total clockwise path length, namely

$$w_1+\cdots+w_n=kd,$$

where $w_i$ is the clockwise distance from the block containing $i$ to the block containing $(i+1)$, with indices taken modulo $n$.

Early's conjecture. The number of hypersimplicial decorated ordered set partitions of type $(k,n)$ with winding number $d$ is $h^{*}_d(\Delta_{k,n})$.

This gives a combinatorial interpretation of the Ehrhart $h^{*}$-vector of the hypersimplex $\Delta_{k,n}$. The supplied source does not state whether the conjecture has been resolved.