Spectral transition conjecture for censored additive spiked matrices

Let $\bm{x}$ be random as in the paper's spiked-prior assumption, let $p(y)$ be a density defining an additive noise channel $\mathcal{P}$, and let $F_{\mathcal{P}}$ denote its Fisher information. Fix $\eta\in(0,1)$ and define the local Fisher score $f(y)=-p'(y)/p(y)$, applied entrywise to matrices. Let

$$\bm{Y}^{(0)}=\frac{\lambda}{\sqrt n}\bm{x}\bm{x}^{\top}+\bm{W},$$

where $W_{ij}=W_{ji}\sim\mathcal{P}_0$ are i.i.d. with density $p$ and $W_{ii}=0$, and form $\bm{Y}$ by independently replacing each upper-triangular entry of $\bm{Y}^{(0)}$ by zero with probability $\eta$, making the same replacement symmetrically below the diagonal. Censored spectral transition conjecture. There exists $\gamma\in\mathbb{R}$ depending only on $p$ and $\eta$ such that: (i) if $\lambda<1/\sqrt{(1-\eta)F_{\mathcal{P}}}$, then $n^{-1/2}\lambda_{\max}(f(\bm{Y}))\to\gamma$ in probability; (ii) if $\lambda>1/\sqrt{(1-\eta)F_{\mathcal{P}}}$, then for some $\varepsilon=\varepsilon(\lambda)>0$, $n^{-1/2}\lambda_{\max}(f(\bm{Y}))\to\gamma+\varepsilon$ in probability. This predicts that entrywise Fisher-score preprocessing yields a spectral algorithm whose transition matches the computational threshold suggested by the censored channel's Fisher information. The source says the conjecture is numerically verifiable but non-trivial to prove using standard random-matrix techniques because censoring corrupts the additive structure.