Second-largest eigenvalue conjecture for balanced complete-graph configurations

Let $d\geq 3$, and let $p:[n]\to{\mathbb R}^d$ satisfy $\|p(i)\|=1$ for every $i\in[n]$ and

$$\sum_{i=1}^n p(i)=0.$$

Assume that the image of $p$ has size at least $3$. Second-largest eigenvalue conjecture. The second largest eigenvalue of $L(K_n,p)$ is $n/2$, and its multiplicity is exactly $n-1$. This conjecture concerns the spectrum of the Laplacian associated with balanced point configurations; the preceding argument establishes that $n/2$ has multiplicity at least $n-1$, while the asserted second-largest-eigenvalue and exact-multiplicity statement remains open.