Planted Jensen-gap conjecture for the Random Subsequence Model

Let $f_{\mathrm{pl}}(\alpha)$ and $f_{\mathrm{pl}}^{\mathrm{ann}}(\alpha)$ denote the planted quenched and annealed free energies, respectively, for aspect ratio $\alpha$. Planted Jensen-gap conjecture. For every $\alpha\in(0,1)$,

$$f_{\mathrm{pl}}(\alpha)<f_{\mathrm{pl}}^{\mathrm{ann}}(\alpha).$$

This conjecture asserts that the Jensen gap for the planted Random Subsequence Model is nontrivial throughout the interior of the parameter range, analogously to the corresponding result for the null model. The paper presents it as a conjectural analogue of the established annealed result; its resolution is left open.