  Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped
with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian
manifold, and in this geometric setting, higher Laplacians $L_m$ are defined
for each positive integer $m$, which generalize the ordinary Laplace-Beltrami
operator. We show that $L_1,L_3,\ldots, L_{2r-1}$ form a set of algebraically
independent generators for the algebra $\mathbb{D}_G(X)$ of $G$-invariant
differential operators on $X$, where $r$ denotes the rank of $X$. This confirms
a conjecture of Engli\v{s} and Peetre, originally stated for the class of
Hermitian symmetric spaces.
