Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with a minuscule node $r$

We prove the perfectness of Kirillov-Reshetikhin crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with $r$ being the minuscule node and $s\ge 1$ using the polytope model of KR crystals introduced by Jang.