  We study support recovery for a $k \times k$ principal submatrix with
elevated mean $\lambda/N$, hidden in an $N\times N$ symmetric mean zero
Gaussian matrix. Here $\lambda>0$ is a universal constant, and we assume $k = N
\rho$ for some constant $\rho \in (0,1)$. We establish that {there exists a
constant $C>0$ such that} the MLE recovers a constant proportion of the hidden
submatrix if $\lambda {\geq C} \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}}$,
{while such recovery is information theoretically impossible if $\lambda = o(
\sqrt{\frac{1}{\rho} \log \frac{1}{\rho}} )$}. The MLE is computationally
intractable in general, and in fact, for $\rho>0$ sufficiently small, this
problem is conjectured to exhibit a \emph{statistical-computational gap}. To
provide rigorous evidence for this, we study the likelihood landscape for this
problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}{\rho}
\log \frac{1}{\rho} } \ll \lambda \ll \frac{1}{\rho^{1/2 + \varepsilon}}$, the
problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a
direct consequence, we establish that a family of local MCMC based algorithms
do not achieve optimal recovery. Finally, we establish that for $\lambda >
1/\rho$, a simple spectral method recovers a constant proportion of the hidden
submatrix.
