Uniqueness of the maximal ideal of operators on the ${\mathrm{\ell }}_p$-sum of ${\mathrm{\ell }}_{\mathrm{\infty }}^n(n\in \mathbb{N})$ for $1<p<\mathrm{\infty }$

A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathcal{B}(X)$ of bounded, linear operators on the Banach space $X=({⨁}_{n\in \mathbb{N}}{\mathrm{\ell }}_{\mathrm{\infty }}^n{)}_{{\mathrm{\ell }}_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=({⨁}_{n\in \mathbb{N}}{\mathrm{\ell }}_{\mathrm{\infty }}^n{)}_{{\mathrm{\ell }}_p}$ and $X=({⨁}_{n\in \mathbb{N}}{\mathrm{\ell }}_1^n{)}_{{\mathrm{\ell }}_p}$ whenever $p\in (1,\mathrm{\infty })$.