Large-eigenvalue sum conjecture for complete-graph Laplacians

Let $p:[n]\to{\mathbb R}^d$ be injective. Large-eigenvalue sum conjecture. The sum of the $n$ largest eigenvalues of $L(K_n,p)$ is at least

$$\frac{n(n+1)}{2}.$$

The conjecture improves the previously established lower bound for this eigenvalue sum. Since the largest eigenvalue is $n$ for every non-constant $p$, it is equivalently a lower bound of $n/2$ for the average of the next $n-1$ largest eigenvalues; it remains open.