Polymatroid monotonicity conjecture for matroid h*-vectors

Let $E$ be a finite set, let $\mathrm{P}_1$ and $\mathrm{P}_2$ be polymatroids on $E$, and let $B(\mathrm{P}_1)$ and $B(\mathrm{P}_2)$ denote their base polytopes. Suppose that

$$B(\mathrm{P}_1)\subseteq B(\mathrm{P}_2).$$

For a loopless matroid $\mathrm{M}$ on $E$, let $h^*_i(\mathrm{M},\underline{\mathcal{L}}_{B(\mathrm{P})})$ be the $i$th coefficient of the $h^*$-vector associated with the line bundle $\underline{\mathcal{L}}_{B(\mathrm{P})}$. Polymatroid monotonicity conjecture. For every loopless matroid $\mathrm{M}$ and every $i$,

$$h^*_i(\mathrm{M},\underline{\mathcal{L}}_{B(\mathrm{P}_1)})\leq h^*_i(\mathrm{M},\underline{\mathcal{L}}_{B(\mathrm{P}_2)}).$$

This is inspired by Stanley's monotonicity result for $h^*$-vectors of polytopes and specializes to the corresponding conjecture for the Boolean matroid. Its validity for general loopless matroids remains open.