Degree-one dominance conjecture for analytic-rank-two newforms

Consider weight $2$ newforms, ordered by level, and call a newform’s degree the degree of its rationality field over $\mathbb Q$. Degree-one analytic-rank conjecture. Among all weight $2$ newforms with analytic rank at least $2$, $100\%$ have degree $1$ in the ordering by level. The statement is motivated by the relation between nontrivial zeroes and analytic rank of special base changes, together with the smaller-degree heuristic; the source gives no proof or resolution.