The truncate-and-relax algorithm's threshold conjecture for hypergraph community recovery

Let $\mathcal{H}$ be the hypergraph stochastic block model with community assignment $\sigma$, let $\widehat{\Sigma}(\mathcal{H})$ denote the semidefinite-programming estimator, and let $I_2(\alpha,\beta)$ be the threshold function defined by

$$I_2(\alpha,\beta) = \max_{t\geq 0} \frac{1}{2^{k-1}}\left[ \alpha (1-e^{-(k-1)t}) + \sum_{a=1}^{k-1} \beta \binom{k-1}{a} \left(1-e^{-(k-1-2a)t}\right)\right].$$

The truncate-and-relax threshold conjecture. If $I_2(\alpha,\beta) > 1$, then

$$\widehat{\Sigma}(\mathcal{H}) = \sigma\sigma^T$$

with probability $1-o_n(1)$. The conjecture proposes that $I_2(\alpha,\beta)=1$ is the correct performance threshold for the truncate-and-relax algorithm; the cited theorem establishes failure below the threshold and success of the truncated estimator above it, but does not establish success of the full semidefinite-programming estimator.