Quaternionic eigenform degree conjecture for nontrivial zeroes

Let $\varphi\in S(\mathcal O_N)$ be an eigenform, and let $K_\varphi$ be the number field generated by its Hecke eigenvalues. Define the degree of $\varphi$ to be $[K_\varphi:\mathbb Q]$. A zero is nontrivial if it is not forced by the local signs. Quaternionic eigenform degree conjecture. Quaternionic eigenforms of smaller degree are more likely to have nontrivial zeroes. The claim is presented as a rough heuristic form; the paper gives computational evidence and notes that a more precise formulation is stated later, but no resolution is supplied.