Chen–Gonzales–Goodman–Kang–Okoudjou conjecture on minimizers of the p-frame potential

Let $d\geq 2$. For $k\in\{1,2,\ldots,d\}$, let $L_k^d$ be the set of $d+1$ unit vectors in $\mathbb{R}^d$ whose absolute pairwise inner products satisfy

$$\lvert\langle {\mathbf x}_i,{\mathbf x}_j\rangle\rvert=\begin{cases}\frac{1}{k},&i,j\in\{1,\ldots,k+1\},\ i\ne j,\\1,&i=j,\\0,&\text{otherwise},\end{cases}$$

and define

$$p_0:=0,\qquad p_d:=2,\qquad p_k:=\frac{\ln(k+2)-\ln(k)}{\ln(k+1)-\ln(k)}\quad\text{for }k\in\{1,2,\ldots,d-1\}.$$

Chen–Gonzales–Goodman–Kang–Okoudjou's conjecture. If $p\in(p_{k-1},p_k]$ for $k=1,2,\ldots,d$, then $L_k^d$ minimizes the $p$-frame potential when $N=d+1$.

This conjecture concerns the transition between different lifted equiangular tight frames as minimizers of the $p$-frame potential for $0<p<2$. The case $d=2$ is known, while the conjecture remains open for $d>2$.