SBP algorithmic threshold conjecture

Let $G\nin\mathbb{R}^{m\times n}$ have independent standard-normal entries. For constraint density $\alpha=\lim_{n\to\infty}m/n$ and $\kappa\in\mathbb{R}_+$, consider the statistical SBP $\mathbf{\mathcal S}(G,\kappa,\alpha)$. Define its algorithmic threshold by

$$\alpha_a(\kappa)=\max\left\{\alpha\ \middle|\ \lim_{n\to\infty}\mathbb{P}\bigl(\mathbf{\mathcal S}(G,\kappa,\alpha)\text{ is solvable in polynomial time}\bigr)=1\right\}.$$

Let $\bar{\psi}_{rd}^{(r)}=\bar{\psi}_{rd}$, with $\bar{\psi}_{rd}$ and $\hat{\mathbf p},\hat{\mathbf q},\hat{\mathbf c},\hat{\gamma}_{sq}$ as in the cited definitions and theorem. SBP algorithmic threshold conjecture. There is a non-increasing convergent sequence $\alpha_c^{(r)}(\kappa)$ defined by

$$\alpha_c^{(r)}(\kappa)=\left\{\alpha\ \middle|\ \bar{\psi}_{rd}^{(r)}(\hat{\mathbf p},\hat{\mathbf q},\hat{\mathbf c},\hat{\gamma}_{sq})=0\right\},$$

such that

$$\alpha_c^{(r)}(\kappa)\geq\alpha_a(\kappa),\qquad \lim_{r\to\infty}\alpha_c^{(r)}(\kappa)=\hat{\alpha}_a(\kappa)=\alpha_a(\kappa).$$

Since $\alpha_c(\kappa)=\alpha_c^{(2)}(\kappa)$, the statistical computational gap is

$$SCG=\alpha_c(\kappa)-\alpha_a(\kappa)=\alpha_c^{(2)}(\kappa)-\lim_{r\to\infty}\alpha_c^{(r)}(\kappa).$$

The claim proposes that the fl-RDT hierarchy converges to the SBP algorithmic threshold, with the finite-$r$ quantities providing upper estimates; the source gives no resolution, so the conjecture remains open.