Positive-even-rank dimension conjecture for prime level newforms

For each prime $p$, let $\operatorname{perdim}(p)$ be the number of weight-$2$ newforms $f$ for $\Gamma_0(p)$ such that $w(f)=-f$ and $L(f,1)=0$. Positive-even-rank dimension conjecture. The following two forms are proposed:

1. Weak form:

$$\lim_{p\to\infty}\frac{\operatorname{perdim}(p)}{\log p}=0$$

as $p$ runs through all prime numbers.

2. Strong form: $\operatorname{perdim}(p)$ is uniformly bounded as $p$ runs through all prime numbers.

The conjecture is supported by the reported computations and by heuristics involving Galois orbits and analytic ranks; neither form is resolved in the supplied text.