The spectral extremal conjecture for bowties with q added edges

Let $T_{n,2}$ be the balanced complete bipartite graph on $n$ vertices, and let $Y_{n,2,q}$ be obtained from $T_{n,2}$ by embedding $q$ pairwise disjoint edges into the vertex part of size $\lceil n/2\rceil$. Let $\lambda(G)$ denote the spectral radius, and let a bowtie be the graph consisting of two triangles sharing one vertex.

Spectral bowtie conjecture. For $q\ge 3$, if $G$ is a graph of sufficiently large order $n$ satisfying

$$\lambda(G)\ge\lambda(Y_{n,2,q}),$$

then $G$ has at least

$${q\choose 2}\left\lfloor\frac{n}{2}\right\rfloor$$

bowties, and $Y_{n,2,q}$ is the unique spectral extremal graph.

This proposes a spectral analogue of the corresponding edge-extremal result for bowties when $q=2$, extending it to arbitrary fixed $q\ge3$. The asserted uniqueness and the general spectral threshold remain open in the source.