Jiang et al.'s forbidden-subgraph multiplicity conjecture

Let $\lambda>0$ and $p\in\mathbb{N}^+$. For a finite family $\mathcal{F}$ of signed graphs, let $M_{p,\mathcal{F}}(\lambda,N)$ be the maximum of $\operatorname{mult}(-\lambda,G)$ over all $p$-colorable signed graphs $G$ on at most $N$ vertices that contain no member of $\mathcal{F}$ as an induced subgraph and satisfy $\lambda_{p+1}(G)\geq-\lambda$, where $\lambda_{p+1}(G)$ is the $(p+1)$-st smallest eigenvalue of $G$. Jiang et al.'s forbidden-subgraph conjecture. For every $\lambda>0$ and $p\in\mathbb{N}^+$, there exists a finite family $\mathcal{F}$ of signed graphs such that $\lambda_1(F)<-\lambda$ for every $F\in\mathcal{F}$ and

$$M_{p,\mathcal{F}}(\lambda,N)\leq \begin{cases} N/k_p(\lambda)+o(N) & \text{if }k_p(\lambda)<\infty,\\ o(N) & \text{otherwise}. \end{cases}$$

This conjecture is the forbidden-subgraph estimate underlying the proposed asymptotic formula for $N_{\alpha,\beta}(d)$. The supplied context does not report a resolution, so the general assertion remains open.