Mean-field and rank-one free-energy comparison conjecture

For $\alpha\in[0,1]$, let $f_{\mathrm{null}}(\alpha)$ be the quenched free energy of the null Random Subsequence Model, let $f^{\mathrm{BMM}}(\alpha)$ be the quenched free energy of the Bernoulli Matching Model, and let $f^{\mathrm{SW}(a,b)}(\alpha)$ be the quenched free energy of the Strict-Weak lattice polymer with parameters $(a,b)$. Mean-field versus rank-one free-energy conjecture. For all $\alpha\in[0,1]$,

$$f_{\mathrm{null}}(\alpha)\leq f^{\mathrm{BMM}}(\alpha)\leq f^{\mathrm{SW}(1,1/2)}(\alpha).$$

This conjecture compares the null Random Subsequence Model with two mean-field models: the Bernoulli Matching Model and the Strict-Weak lattice polymer. The paper poses the comparison as an open problem; the following open problem suggests pursuing an analogous relationship for the planted setting.