Li–Feng–Peng's spectral supersaturation conjecture for the bowtie

Let $F_2$ be the bowtie graph, let $\tau(G)$ denote the number of copies of $F_2$ in a graph $G$, and let $K_{s,t}^{q}$ be obtained from $K_{s,t}$ by adding $q$ pairwise disjoint edges within the part of size $s$. Write $\binom{q}{2}$ for the number of pairs among these added edges. Li–Feng–Peng's conjecture. If $q\geq 2$ and $G$ is a graph with large order $n$ satisfying

$$\lambda(G)\geq\lambda\left(K_{\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor}^{q}\right),$$

then

$$\tau(G)\geq\binom{q}{2}\left\lfloor\frac{n}{2}\right\rfloor,$$

with equality if and only if $G=K_{\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor}^{q}$. This conjecture concerns spectral supersaturation for the bowtie; the supplied text does not state whether it has been resolved.