Whole-center conjecture for the $\mathrm{SL}_d$-skein algebra

Let $S$ be an oriented surface of finite topological type. For each puncture, let $[P_i]$ denote the skein represented by a small loop around that puncture with weight $i\in\{1,2,\dots,d-1\}$. Let $L^{[\widehat P_d^{(n,i)}]}$ denote the skein obtained by threading the reduced power elementary polynomial $\widehat P_d^{(n,i)}$ around a framed knot $L\subset S\times[0,1]$. Whole-center conjecture. If the $d$-root $q^{\frac1d}$ is chosen so that $q^{\frac2d}$ is a primitive $n$-th root of unity, then the center of $\mathcal S^q_{\mathrm{SL}_d}(S)$ is generated as a subalgebra by the puncture skeins $[P_i]$ and the skeins $L^{[\widehat P_d^{(n,i)}]}$. The center contains the puncture skeins for every value of $q$; the conjecture is proved when $d=2$, while the asserted description for general $d$ remains open.