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

Abstract: A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^{n\bigr)_{\ell_1}$} contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^{n\bigr)_{\ell_p}$} and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^{n\bigr)_{\ell_p}$} whenever $p\in(1,\infty)$.