Spectral stability conjecture for minimum $Q$-degree hypergraphs

Let $p>1$, $s\geq r\geq2$, let $Q$ be an $s$-vertex $r$-graph, and let $\mathcal{F}$ be a family of $r$-graphs with $\pi(Q,\overline{\mathcal{F}})>0$. For $n$ vertices, let $\mathcal{G}_{n}$ be the collection of all $n$-vertex $\mathcal{F}$-free $r$-graphs with minimum $Q$-degree at least

$$(1-\varepsilon)\pi(Q,\overline{\mathcal{F}})\binom{n}{s-1},$$

and define

$$\lambda^{(p)}(Q,\mathcal{G}_{n})=\max\{\lambda^{(p)}(Q,G):G\in\mathcal{G}_{n}\}.$$

The spectral stability conjecture. There exists $n_{0}$ such that for any $\mathcal{F}$-free $r$-graph $H$ on $n\geq n_{0}$ vertices,

$$\lambda^{(p)}(Q,H)\leq\lambda^{(p)}(Q,\mathcal{G}_{n}).$$

Moreover, if equality holds, then $H\in\mathcal{G}_{n}$. The paper presents this as a proposed extension of its spectral stability theorem, obtained by removing an additional minimum-degree condition; no resolution is supplied in the given text.