Radicality conjecture for off-diagonal entries of Gram matrices

Let $K_n$ be the complete graph on $n$ vertices, let $\mathbb{k}$ be a field, and let $\operatorname{asym}_{\mathbb{k}}(\mathrm{radical},K_n)$ denote the asymptotic threshold for radicality of the corresponding Lovász–Saks–Schrijver ideals. Equivalently, let $X$ be an $n\times d$ matrix of variables, and consider the ideal generated by the off-diagonal entries of $XX^T$.

Radicality conjecture.

$$\operatorname{asym}_{\mathbb{k}}(\mathrm{radical},K_n)=1$$

(at least if $\operatorname{char}(\mathbb{k})=0$); equivalently, the ideal of the off-diagonal entries of $XX^T$ is radical for all $n$ and $d$.

This conjecture gives a concrete radicality prediction for the Lovász–Saks–Schrijver ideal associated with the complete graph. The source specifies the characteristic-zero case, and no resolution is provided.