General spectral supersaturation conjecture for friendship graphs

For an integer $k\geq 1$, define

$$f(k)= \begin{cases} k^2-k,&\text{if }k\text{ is odd};\\ k^2-\frac{3}{2}k,&\text{if }k\text{ is even}. \end{cases}$$

Let $F_k$ be the friendship graph and let $\tau_k(G)$ denote the number of copies of $F_k$ in $G$. Let $K_{s,t}^{q}$ be obtained from $K_{s,t}$ by adding $q$ pairwise disjoint edges within the part of size $s$. The general spectral supersaturation conjecture. For every $k\geq 1$, there exists an absolute constant $\delta>0$ such that, for any sufficiently large $n$ and $f(k)+1\leq q\leq\delta\sqrt n$, if $G$ is an $n$-vertex graph satisfying

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

then

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

with equality if and only if $G=K_{\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor}^{q}$. This is presented as a conjectural extension of known extremal results, and the supplied text gives no resolution.