Phase transition conjecture for PCA subspace and score recovery

Let $n,d\to\infty$ with $n/d\to c>0$ and $n_\ell/n\to p_\ell$ for $\ell\in\{1,\dots,L\}$. Let $A$ and $\beta_i$ be the quantities defined in the paper, and let $\hat{u}_i$ be the $i$th principal component and $\hat{z}^{(i)}/\sqrt n$ the corresponding normalized score vector. Phase transition conjecture. If $A(\beta_i)\leq 0$, then

$$\left|\left\langle \hat{u}_i,\operatorname*{Span}\{u_1,\dots,u_k\}\right\rangle\right|^2\overset{a.s.}{\longrightarrow}0, \qquad \left|\left\langle \frac{\hat{z}^{(i)}}{\sqrt n},\operatorname*{Span}\{z^{(1)},\dots,z^{(k)}\}\right\rangle\right|^2\overset{a.s.}{\longrightarrow}0.$$

This predicts that both principal-component and score-vector recovery vanish at the phase transition $A(\beta_i)=0$, extending the positive-recovery formulas to the regime where $A(\beta_i)\leq0$.