Replica-symmetry MP-structure conjecture for the independent set problem

Let $(\beta,h,\gamma)$ be parameters of the independent set problem, and let an MP-structure consist of a hierarchy $(\zeta,\mathcal V_{\boldsymbol m})$ with $K$ levels. For such a structure, write $R_{N,\beta,h,\gamma}(\mathcal S)$ for the rest-term in the interpolation formula, where $\mathcal S=(\zeta,\mathcal V_{\boldsymbol m})$. Replica-symmetry MP-structure conjecture. For every $(\beta,h,\gamma)$ there exists a unique $K^\star$-level MP-structure $\mathcal S^\star=\mathcal S^\star(\beta,h,\gamma)=(\zeta^\star,\mathcal V^\star_{\boldsymbol m})$ such that

$$\lim_{N\to\infty}R_{N,\beta,h,\gamma}(\mathcal S^\star)=0.$$

The case $K^\star=\infty$ is also possible. This conjecture asserts that the limiting free energy is captured by a unique hierarchical MP-structure through the vanishing of the interpolation remainder, which is the key step in describing the replica-symmetry phase of the independent set problem.