Repeated orthonormal sequence threshold conjecture for p-frame energy

Let $N=m+kd$ points be given in $\mathbb{S}^{d-1}$, where $1\leq m<d$ and $d\geq 2$, and let $\mathbf{A}\in\mathbb{R}^{N\times N}$ be their Gram matrix. The repeated orthonormal sequence is $\{e_{j\bmod d}\}_{j=1}^N$, and its energy is $d(k^2-k)+2k$. Repeated orthonormal sequence threshold conjecture. There is a value $p_0$, independent of $d$ and $m$, such that the repeated orthonormal sequence minimizes $E_p$ over all systems of $N$ unit vectors for $p<p_0$, while the minimum satisfies $E_p(\mathbf{A})<d(k^2-k)+2k$ for $p>p_0$. Moreover, $p_0=p_0(k)$ satisfies $p_0(k)\to2$ as $k\to\infty$.