Conjecture on Grammians of independent symmetric rank-one tensors

Let $a_1,\ldots,a_K$ be independent copies of the random vector $a \sim \operatorname{Unif}(\mathbb{S}^{D-1})$, and let $G_n$ be the Grammian associated with these vectors, with entries $(G_n)_{ij}=\langle a_i,a_j\rangle^n$. Fix $\epsilon>0$ arbitrarily.

Grammians conjecture. There exists a constant $\kappa_n>0$ and an increasing function $\gamma_n:(0,\kappa_n)\to(0,1)$, both depending only on $n$, such that $\gamma_n(\kappa)\to1$ as $\kappa\to0$, and whenever $K\leq\kappa D^n$ for some $\kappa<\kappa_n$,

$$\mathbb{P}\left(\left\|G_n^{-1}\right\|_2\geq\gamma_n(\kappa)-\epsilon\right)\to1\quad\textrm{as}\quad D\to\infty.$$

In particular, if $K=o(D^n)$, then $\|G_n^{-1}\|_2\to1$ as $D\to\infty$. The conjecture concerns the well-posedness of inverting the Grammian for independent random symmetric rank-one tensors; the paper gives numerical evidence and notes that the stated proof is currently available only for tensor order $n=2$, while the extension to $n>2$ remains open.