Critical tensor covariance at the Marchenko-Pastur threshold

For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \to λ\in (0,\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \ex