Solus's Ehrhart positivity conjecture for base-$r$ simplices

For a positive integer $r$ and a positive integer dimension $d$, define $\boldsymbol{q}_r=(r-1,(r-1)r,(r-1)r^2,\dots,(r-1)r^{d-1})$ and let the base-$r$ $d$-simplex be

$$\mathcal{B}_{(r,d)}:=\Delta_{(1,\boldsymbol{q}_r)}.$$

Here $\Delta_{(1,\boldsymbol{q}_r)}$ is the simplex introduced earlier in the paper. Solus's conjecture. The base-$r$ $d$-simplex is Ehrhart positive; that is, all coefficients of its Ehrhart polynomial are positive. Solus introduced this family in connection with numeral systems and proved that its $h^*$-polynomial is real-rooted. The stated Ehrhart-positivity claim is based on computational evidence.