Monotonicity conjecture for the number of spectral components

Let $F_t$ be the limiting spectral distribution of the noisy sample covariance matrix at noise power $t\geq 0$, and let $N(t)$ denote the number of connected components of its support. Under the standing assumptions of the paper, $N(t)\in\{1,2\}$ for all $t\geq 0$. Monotonicity conjecture. $N$ is non-increasing in $t$. This would imply that once the separation between the information and noise eigenvalues disappears, it does not reappear as the noise power increases; the paper gives simulations and notes analogous results for other random matrix models, but does not establish the claim.