Growth conjecture for nontrivial zeroes of quaternionic modular forms

For $N\in\mathrm{Sq}_{\mathrm{odd}}$, choose a maximal quaternionic order $\mathcal O_N$ of level $N$ in $B_N$, and let a nontrivial zero be a zero of an eigenform in $S(\mathcal O_N)$ that is not forced by local sign conditions. Nontrivial zero growth conjecture. As $N$ tends to infinity along $\mathrm{Sq}_r$ for some $r$, or along $\mathrm{Sq}_{\mathrm{odd}}$, the number of nontrivial zeroes for $S(\mathcal O_N)$ is

$$O(N^{1+\epsilon})$$

for every $\epsilon>0$. The prediction is intended to quantify the expectation that most zeroes are trivial; the paper supplies heuristic motivation but no proof or resolution.