Universal optimality conjecture for lifted ETFs in the p-frame potential

Let $d\geq 2$. For each natural number $k$ with $1\leq k\leq d-1$, define

$$p_k=\frac{\log(k+2)-\log k}{\log(k+1)-\log k}.$$

Also define $p_0=0$. The configurations $L_k^d$ are the lifted ETF configurations in dimension $d$, and $\operatorname{FP}_{p,d+1,d}$ denotes the $p$-frame potential for $d+1$ vectors in $\mathbb{R}^d$.

Universal optimality conjecture for lifted ETFs. The following configurations minimize the $p$-frame potential $\operatorname{FP}_{p,d+1,d}$: when $p\in(p_{k-1},p_k]$, the $L_k^d$ configuration for $k=1,2,\ldots,d-1$; and when $p\in(p_{d-1},\infty]$, the $\operatorname{ETF}_d$, or equivalently the $L_d^d$, configuration.

This conjecture proposes a complete sequence of minimizers as the parameter $p$ varies, explaining the numerical evidence that lifted ETFs are optimal configurations for the $p$-frame potential. The supplied text gives no resolution, so its status remains open.