  We are concerned in this article with a classical question in spectral
geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the
constancy of holomorphic sectional curvature of a complex $n$-dimensional
compact K\"ahler manifold can be completely determined by the eigenvalues of
its $p$-Laplacian for a \emph{single} integer $p$? We treat this question in
this article under two Einstein-type conditions: cohomologically Einstein and
Fano Einstein. Building on our previous work, we show that for cohomologically
Einstein K\"ahler manifolds this is true for all but finitely many pairs
$(p,n)$. As a consequence, the standard complex projective spaces can be
characterized among cohomologically Einstein K\"ahler manifolds in terms of a
single spectral set in all these cases. Moreover, in the case of $p=0$, we show
that the complex projective spaces can be characterized among Fano
K\"ahler-Einstein manifolds only in terms of the first nonzero eigenvalue with
multiplicity, which has a similar flavor to a recent remarkable result due to
Kento Fujita.
