The $q$-moment conjecture for isotropic probability masses

Let $k$ be a positive integer, set

$$\beta=\frac{1}{\sqrt{k+2}},\qquad N=\binom{k+1}{2},$$

and let an isotropic probability mass $\mu$ on $\mathbb{R}^k$ be a probability mass with the isotropy property used in the paper. For $1\leq q\leq2$, consider independent $x,y\sim\mu$.

The $q$-moment conjecture. If $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, then

$$\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|^q\leq\beta^q+\frac{1-\beta^q}{N},$$

with equality if and only if there is $X\subseteq\mathbb{R}^k$, a system of $N$ equiangular lines, such that

$$\mu(x)+\mu(-x)=\frac1N$$

for every $x\in X$. This conjecture would give sharp moment bounds relevant to lower bounds for $L^p$ norms of off-diagonal matrix entries. Its status was subsequently resolved by Glazyrin, so it is recorded as solved rather than open.