Engliš–Peetre conjecture on higher Laplacians for Hermitian symmetric spaces

Let $X=G/K$ be a simple Hermitian symmetric space, where $K$ is a compact subgroup of $G$. For each $m\in\mathbb{N}$, let $L_m$ denote the higher Laplacian of order $2m$ on $X$, with $L_1$ the Laplace–Beltrami operator. Engliš–Peetre conjecture. The operators $L_m$, $m\in\mathbb{N}$, generate the whole algebra $\mathbb{D}_G(X)$ of $G$-invariant differential operators on $X$. The conjecture proposes a geometrically defined generating set for invariant differential operators on Hermitian symmetric spaces; the supplied text gives no indication that it has been resolved.