Minimax risk prediction of phase transitions in convex programs

Let $\vct{x}_0\in\mathbb{R}^d$ be the target vector, let $R_{\mathrm{mm}}(\vct{x}_0)$ denote the minimax mean-square-error risk, and consider recovery by the convex program with measurement matrix $\mtx{A}\in\mathbb{R}^{m\times d}$. Suppose that $\mtx{A}$ has independent standard normal entries and that $f:\mathbb{R}^d\to\mathbb{R}$ is convex. Minimax risk predicts phase transitions.

$$m \geq R_{\mathrm{mm}}(\vct{x}_0)+o(d)\quad\Longrightarrow\quad\text{the recovery program succeeds with probability }1-o(1),$$ $$m \leq R_{\mathrm{mm}}(\vct{x}_0)+o(d)\quad\Longrightarrow\quad\text{the recovery program succeeds with probability }o(1).$$

The order notation is intended heuristically. The conjecture asserts that the minimax risk predicts the sharp transition in the number of Gaussian measurements for several convex regularizers, but the statement is presented without a definitive resolution here.