Stability conjecture for distinguishing newforms at squarefree level

Let $n_0(N,k)$ be the smallest nonnegative integer such that, for any newforms $f,g\in S_k(\Gamma_0(N))$, equality $a_n(f)=a_n(g)$ for all $n\leq n_0(N,k)$ implies $f=g$. Let $N\in\mathbb{Z}_{>0}$ be squarefree. Stability conjecture. There exists $K\in\mathbb{Z}_{>0}$ such that if $k\geq K$ is an even integer, then $n_0(N,k)$ is equal to the least prime that does not divide $N$. This predicts that, for fixed squarefree level, the number of initial Fourier coefficients needed to distinguish newforms eventually stabilizes at the first prime not dividing the level.