Strangeness-dimension conjecture for prime level newforms

For each prime $p$, let $\operatorname{strdim}(p)$ denote the strangeness dimension defined in the surrounding discussion. Strangeness-dimension conjecture. The following two forms are proposed:

1. Weak form:

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

as $p$ runs through all prime numbers.

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

The conjecture is motivated by analytic-rank heuristics, Galois-orbit considerations and computational data; neither form is resolved in the supplied text.