Extremal configuration conjecture for $c(d,d+1,d+1)$

Let $c(d,n,k)$ denote the extremal signed-sum quantity studied in the paper, and let $\omega_{d+1}$ be the corresponding extremal vector configuration. For a linear subspace $H\subseteq\mathbb{R}^d$, write $H^\perp$ for its orthogonal complement. Extremal configuration conjecture. For every $d\geqslant1$,

$$c(d,d+1,d+1)=\sqrt{d+2}.$$

Moreover, equality is attained if and only if, up to sign changes, $\omega_{d+1}$ is the union of the vertex set of a regular simplex in an even-dimensional linear subspace $H$, centered at the origin, and an orthonormal basis of $H^\perp$. The source describes this as a conjecture already proved for $d=2$; the general assertion remains open.