Spaces with the maximal projection constant revisited

Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (that is, attaining the classical Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (so that $d=\frac{n(n+1)}{2}$ in the re