Stability conjecture for Maaß newforms

Let $n_0(q,T)$ be the least integer such that two nonmonomial Maaß newforms of weight $0$, level $q$, and principal character, with Laplacian eigenvalues in $(1/4,T]$ and identical Hecke eigenvalues at every prime $p\leq n_0(q,T)$, are equal. Stability conjecture for Maaß newforms. For a positive odd integer $q$, there exists a positive real $T_0$ such that for all $T\geq T_0$, $n_0(q,T)$ equals the smallest prime not dividing $q$ when $q$ is squarefree, while for nonsquarefree $q$ it equals

$$\max_{\substack{q^*\mid q' \\ q^*>1}}\left\{\min\left\{p\colon \varepsilon_{\mathrm{quad}(q^*)}(p)=-1,\ p\nmid q\right\}\right\},$$

where $q'$ is the largest squarefree integer dividing $q$ such that every prime $p$ dividing $q'$ also satisfies $p^2\mid q$, and $\varepsilon_{\mathrm{quad}(q^*)}$ is the unique primitive quadratic character modulo $q^*$. The surrounding discussion motivates the conjecture through quadratic twists and a theorem giving positive proportions of indistinguishable pairs; the conjecture itself is unresolved in the source.