The extremal configuration conjecture for d+1 unit vectors

Let $d\geqslant 1$, and let $(u_i)_{i=1}^{d+1}$ be a configuration of unit vectors in $S^{d-1}\subset\mathbb{R}^d$. A sequence of signs is an element $\varepsilon\in\{\pm1\}^{d+1}$. A linear subspace $H\subset\mathbb{R}^d$ is even dimensional when $\dim H$ is even.

Extremal configuration conjecture. For every such configuration, there exists $\varepsilon\in\{\pm1\}^{d+1}$ such that

$$\left|\sum_{i=1}^{d+1}\varepsilon_i u_i\right|\geqslant\sqrt{d+2}.$$

Moreover, equality in the sharp estimate occurs if and only if, up to sign changes, $(u_i)_{i=1}^{d+1}$ is the union of the vertex set of a regular simplex centered at the origin in an even-dimensional linear subspace $H$ and an orthonormal basis of $H^\perp$. This conjecture concerns the extremal point configurations for the corresponding sign-maximization problem; the source notes that the statement was corrected in the cited literature, while its resolution is not specified here.