Cosine covariance stochastic domination conjecture for Gaussian minima

Let $z\sim\mathcal{N}(0,I_n)$ and, for an $n\times n$ correlation matrix $\Sigma$, define

$$M(\Sigma)=\min_{1\leq i\leq n}|(\Sigma^{1/2}z)_i|.$$

Define the cosine covariance by

$$\Sigma^{\mathrm{cos}}_{ij}=\cos\left(\frac{\pi(i-j)}{n}\right).$$

Cosine covariance conjecture. For every $n\geq 1$, every correlation matrix $\Sigma$, and every $t\geq 0$,

$$\mathbb{P}[M(\Sigma^{\mathrm{cos}})\geq t]\leq\mathbb{P}[M(\Sigma)\geq t].$$

Thus $M(\Sigma^{\mathrm{cos}})$ should be stochastically dominated by $M(\Sigma)$; in particular, for every $p>0$,

$$\mathbb{E}[M(\Sigma^{\mathrm{cos}})^p]\leq\mathbb{E}[M(\Sigma)^p].$$

The paper presents this as a stronger revision of Litvak's refuted moment conjecture and states that it has not been proved; the proposed stochastic domination remains open.