The p-adic Brauer–Siegel conjecture for p-ramification torsion

Fix a prime $p$ and an infinite family ${\mathcal K}$ of number fields. For each $K\in{\mathcal K}$, let ${\mathcal T}_{K,p}$ denote the relevant finite $p$-ramification torsion group, let $D_K$ be the discriminant of $K$, and let $v_p$ be the $p$-adic valuation.

The p-adic Brauer–Siegel conjecture. There exists a constant ${\mathcal C}_p({\mathcal K})$ such that

$$v_p(\# {\mathcal T}_{K,p})\leq {\mathcal C}_p({\mathcal K})\cdot \frac{\log_\infty(\sqrt{D_K})}{\log_\infty(p)}$$

for all $K\in{\mathcal K}$, where $\log_\infty$ is the usual complex logarithm. The paper presents extensive numerical computations in this direction, while the asserted uniform bound remains open.