Random square DFT submatrix norm conjecture

Let $n$ be a positive integer, let $\mathbf{F}$ be the normalized $n\times n$ DFT matrix, and let $\mathsfsl{P}_\delta$ and $\mathsfsl{P}_\delta'$ denote the coordinate projections selecting random subsets of rows and columns of equal density $\delta$. Quartercircle Law. A random square submatrix of the $n\times n$ DFT satisfies

$$\operatorname{\mathbb{E}}\left\Vert\mathsfsl{P}_\delta\mathbf{F}\mathsfsl{P}_\delta'\right\Vert\leq 2\sqrt{\delta(1-\delta)}.$$

The inequality becomes an equality as $n\to\infty$. This conjecture concerns the limiting behavior of the expected norm when the dimension grows while the proportion of selected rows and columns remains fixed; the source presents it as a direction for further research, and no resolution is given.