Quaternionic modular forms have asymptotically only trivial zeroes at fixed factor count

Let $r$ be odd. For each $N\in\mathrm{Sq}_r$, choose a maximal order $\mathcal O_N$ in the definite quaternion algebra $B_N$ of discriminant $N$. An eigenform in $S(\mathcal O_N)$ has a trivial zero when its value vanishes at an ideal class for a reason forced by its local signs; all other zeroes are nontrivial. Quaternionic zero conjecture. As $N$ ranges over $\mathrm{Sq}_r$ and tends to infinity, $100\%$ of the zeroes of eigenforms are trivial zeroes, and $100\%$ of eigenforms have no trivial zeroes. This conjecture is a heuristic prediction about the distribution of zeroes of quaternionic modular forms; the source provides numerical and heuristic motivation but no proof.