  In this note, we generalize a mean-value inequality of Guo-Phong-Sturm to the
setting of a compact K\"ahler orbifold. This shows that their reasoning is
insensitive to quotient singularities. As we aim for a self-contained
exposition, we generalize some fundamental results, namely the
$\alpha$-invariant estimate by H\"ormander and Tian, an approximation result
for the psh-envelope of a $(1,1)$-form in a K\"ahler class by Berman and an
$L^\infty$ estimate for this envelope by Guo-Phong-Tong-Wang.
