Repeated-point scaling conjecture for the spectral gap

Let $p:[n]\to{\mathbb R}^d$ be injective, let $G$ be a graph with vertex set $[n]$, and write $\lambda(G,p)=\lambda_{\binom{d+1}{2}+1}(L(G,p))$. For $k\geq 1$, let $p^k:[kn]\to{\mathbb R}^d$ satisfy $|(p^k)^{-1}(p(v))|=k$ for every $v\in[n]$. Repeated-point scaling conjecture. For every injective mapping $p:[n]\to{\mathbb R}^d$ and every $k\geq 2$,

$$\lambda(K_{kn},p^k)=\frac{k}{2}\lambda(K_{2n},p^2).$$

The claim predicts linear scaling of the relevant spectral gap when each point is repeated equally. It is motivated by the paper's analysis of repeated points in Turán-graph examples, but remains unproved in the source.