Almost-flat fiducial conjecture for SICs in dimensions n^2+3

Let $d=n^2+3=p$ be a prime in the specified sequence of dimensions, and let $obreak\hat{\Psi}=(x_0,x_1,\ldots,x_{d-1})^{\rm T}$ be an un-normalized SIC fiducial. Almost-flat fiducial conjecture. In each dimension in the sequence, there \exists an un-normalized SIC fiducial $\hat{\Psi}$ such that

$$x_j=\begin{cases}-2-\sqrt{d+1},&j=0,\sqrt{x_0e^{i\vartheta_j}},&j>0,\end{cases}$$

for suitable phases $e^{i\vartheta_j}$; equivalently, $|x_1|=\cdots=|x_{d-1}|$. The conjecture is part of a proposed construction of SICs in the prime dimensions $d=n^2+3$; the source presents it as unproved and motivated by the observed symmetries.