A free fourth moment theorem for fully symmetric chaotic eigenfunctions

Let $\mathcal{M}$ be the tracial von Neumann algebra in the abstract derivation setting, with generator $\Delta$ and carré du champ operator $\Gamma$, and let $X\in L^2(\mathcal{M})$ be self-adjoint and chaotic of order $q\geq 1$. Assume that for every $t\geq 0$ its directional derivative satisfies $\delta_t(X)^*=\delta_t(X)$, and normalize it by $\tau(X^2)=1$. Free fourth moment conjecture. There exists a constant $C_q$, depending only on $q$, such that

$$\left\lVert \Gamma(X)-q.1\otimes1\right\rVert_{L^2(\mathcal{M})\bar{\otimes}L^2(\mathcal{M}^{op})}\leq C_q\sqrt{\tau(X^4)-2}.$$

Such a bound would imply an abstract free fourth moment theorem, namely that convergence of the fourth moment to the semicircular value $2$ forces convergence of the carré du champ to $q.1\otimes1$. The conjecture is presented as a free counterpart of Ledoux's result; the source does not establish it in the abstract setting.