Degree-one dominance conjecture for nontrivial quaternionic zeroes

For $N\in\mathrm{Sq}_{\mathrm{odd}}$, choose a maximal order $\mathcal O_N$ in the definite quaternion algebra $B_N$. For an eigenform $\varphi\in S(\mathcal O_N)$, its degree is the degree of the number field generated by its Hecke eigenvalues; a zero is nontrivial if it is not forced by local sign conditions. Degree-one zero conjecture. As $N$ tends to infinity along $\mathrm{Sq}_r$ for some odd $r$, or along $\mathrm{Sq}_{\mathrm{odd}}$, $100\%$ of the nontrivial zeroes of $S(\mathcal O_N)$ come from degree-$1$ forms. This is the precise version of the paper’s rough degree heuristic and is supported by computational data and Galois-orbit considerations, but remains unproved.