Gamma-to-cumulant conjecture for iterated Malliavin Gamma operators

Let $F_\infty$ be the target random variable in the second Wiener chaos of the form referenced in the source, and let $F=I_q(f)$ be a chaotic random variable in the $q$th Wiener chaos, where $q\geq2$. Let $a_r$, $\Gamma_r$, and $\Delta(F)$ denote the coefficients, iterated Malliavin Gamma operators, and discrepancy quantity used in the estimate. Gamma-to-cumulant conjecture. There exists a constant $C$, possibly depending on $q$ and $d$, such that

$$\operatorname{Var}\left(\sum_{r=1}^{d+1}a_r\Gamma_{r-1}(F)\right)\leq C\Delta(F).$$

In the particular case $d=2$ and $\alpha_{\infty,1}=-\alpha_{\infty,2}=1/2$, so that $F_\infty=N_1N_2$ for independent standard normal variables $N_1,N_2$, this becomes

$$\operatorname{Var}\left(\Gamma_2(F)-F\right)\leq C\left\{\frac{\kappa_6(F)}{5!}-2\frac{\kappa_4(F)}{3!}+\kappa_2(F)\right\}.$$

The conjecture seeks to control the iterated Gamma operators in the Kolmogorov-distance bound using finitely many cumulants; the source suggests that proving the displayed estimate would be a possible route, while no resolution is supplied.