Projected gradient descent convergence conjecture for matrix completion

Let $M^*\in\mathcal{S}^{d\times d}$ be symmetric positive semidefinite, and let $\Omega$ be generated under the model $\operatorname{SMC}(M^*,p)$: each $(i,j)$ with $i\geq j$ is observed independently with probability $p\in(0,1]$, and $(j,i)\in\Omega$ if and only if $(i,j)\in\Omega$. Let $\Pi_{\Omega}$ denote the sampling projection, and let the projected gradient descent algorithm use iterates $M^t$ and step size $\eta_t$. For some numbers $C,C'>0$ depending on the rank $r$ and incoherence parameter $\mu$, the following holds under this model. Projected gradient descent convergence conjecture. If

$$p\geq C\frac{d\log d}{d},$$

then with high probability, projected gradient descent with fixed step size $\eta_t\equiv\eta=\mathcal{O}(1/p)$ outputs a matrix $M^t$ of rank at most $r$ such that

$$\left\|\Pi_{\Omega}(M^t)-\Pi_{\Omega}(M^*)\right\|_{\mathrm{F}}^2\leq\epsilon$$

after $C'\log(1/\epsilon)$ iterations; moreover, $M^t$ converges to $M^*$. This conjecture concerns the empirical convergence behavior of projected gradient descent for matrix completion and remains open according to the source.