Kwan–Wigderson's unbounded inertia conjecture for graphs with independence number two

Let $G=(V,E)$ be a simple graph, and let $\alpha(G)$ denote its independence number. A weighted adjacency matrix of $G$ is a Hermitian matrix $A$ indexed by $V$ such that $A_{ij}\neq 0$ only when $ij\in E$. For a Hermitian matrix $A$, let $n_{\geq 0}(A)$ be the number of nonnegative eigenvalues, and define $n_{\geq 0}(G)$ as the minimum of $n_{\geq 0}(A)$ over all weighted adjacency matrices $A$ of $G$.

Kwan–Wigderson's conjecture. For every integer $k$, there exists a graph $G$ with $\alpha(G)=2$ and

$$n_{\geq 0}(A)\geq k$$

for every weighted adjacency matrix $A$ of $G$.

This conjecture concerns whether the inertia bound $\alpha(G)\leq n_{\geq 0}(G)$ can be arbitrarily far from tight, even among graphs with independence number two. It arose from work on Godsil's question about whether the inertia bound can always be attained by a suitable weighted adjacency matrix; the source provides no resolution of the conjecture.