Refinement conjecture for Levenshtein-type polynomial nodes

Let $q\geq 3$, let $n$ and $\ell$ be fixed, and let $T_n$ denote the discrete set of admissible points used in the relaxed linear programs over $[\ell,s]\cap T_n$ and $[\ell,1]\cap T_n$. For each such choice, let $f_{\rm ref}$ and $g_{\rm ref}$ be the new polynomials constructed from the Levenshtein-type polynomial $f_{2k}^{n,\ell,s}$, whose relevant roots are $\alpha_1,\ldots,\alpha_{k-1}$. Refinement conjecture. For fixed $q\geq 3$, $n$, and $\ell$ there exists a constant $s(q,n,\ell)$ such that whenever $s\in[-1,s(q,n,\ell))\cap T_n$ (that is large enough $d/n=(1-s)/2$), the new polynomials $f_{\rm ref}$ and $g_{\rm ref}$ solve the relaxed linear programming in the context above. Equivalently, for most parameters, the roots $\alpha_1,\alpha_2,\ldots,\alpha_{k-1}$ of the Levenshtein-type polynomial $f_{2k}^{n,\ell,s}(t)$ are the best approximation of the optimal nodes for general linear programming. The conjecture concerns the feasibility and optimality of the refined polynomials in a discrete relaxation of the linear programs; numerical investigations support it in numerous cases, but no resolution is supplied here.