Conjecture on vanishing and nonvanishing traces on Hecke newspaces

Let $m\geq 2$ be a fixed non-square integer, and let $T_m^{\operatorname{new}}(N,k)$ denote the restriction of the $m$-th Hecke operator to the new subspace $S_k^{\operatorname{new}}(\Gamma_0(N))$, where $N$ is coprime to $m$. Newspace trace dichotomy conjecture. There exist integers $N_1$ and $N_2$, both coprime to $m$, such that

$$\operatorname{Tr} T_m^{\operatorname{new}}(N_1,k)=0$$

for all even $k\geq 2$, while

$$\operatorname{Tr} T_m^{\operatorname{new}}(N_2,k)\neq 0$$

for all even $k\geq 2$. The conjecture is based on numerical computations after the authors report that the analogous blanket nonvanishing statement for newspaces is false, with several vanishing families already found. The proposed dichotomy remains open in the supplied source.