Coefficientwise monotonicity conjecture for Ehrhart polynomials under the swap operator

Let $\mathbf{w}$ be a generalized snake word, let $\varphi_i(\mathbf{w})$ be the word obtained by the swap operator, and write $L(\mathbf{w};t)$ for the Ehrhart polynomial associated with $\mathbf{w}$. Coefficientwise monotonicity conjecture. For every $k>0$,

$$[t^k]L(\varphi_i(\mathbf{w});t)\leq[t^k]L(\mathbf{w};t).$$

This conjecture parallels the coefficientwise inequality for the $h^*$-polynomial established earlier in the paper; its general validity remains open.