Two-sided total variation estimate for centered Gamma approximation in the second Wiener chaos

Let $\nu>0$, and let $F=I_2(f)$ belong to the second Wiener chaos with

$$\mathbb{E}[F^2]=2\nu.$$

Let $G(\nu)\sim\operatorname{CenteredGamma}(\nu)$. Two-sided total variation conjecture. There exist constants $0<C_1<C_2$ such that

$$C_1\mathbf{M}(F)\leq d_{TV}(F,G(\nu))\leq C_2\mathbf{M}(F).$$

Here $\mathbf{M}(F)$ is the quantity defined in the paper. The conjecture proposes a sharp two-sided comparison between total variation distance and this second-chaos error quantity for every $\nu>0$.