Exponential critical-ratio conjecture for the symmetrized Frobenius estimator

Let $A$ be an $n\times n$ matrix, and let $M(A)$ be the matrix used to define the symmetrized Frobenius estimator. Write

$$X_{\mathrm{Frob},s}=\|M(A)\|^2$$

and define its critical ratio by

$$\frac{\operatorname{\mathbb{E}}[X_{\mathrm{Frob},s}^2]}{\operatorname{\mathbb{E}}[X_{\mathrm{Frob},s}]^2}.$$

Exponential critical-ratio conjecture. There is a sequence of constants $\theta_d$ with

$$\lim_{d\to\infty}\theta_d=1,$$

such that, for every $n\times n$ matrix $A$,

$$\frac{\operatorname{\mathbb{E}}[X_{\mathrm{Frob},s}^2]}{\operatorname{\mathbb{E}}[X_{\mathrm{Frob},s}]^2}\leq\theta_d^n.$$

This is presented as a stronger conjecture that would imply Barvinok's concentration conjecture via Chebyshev's inequality. Its resolution is not given in the paper.