Regulator and torsion growth conjecture for totally real fields

Let ${\mathcal K}$ be the set of totally real number fields. For $K\in{\mathcal K}$, let $D_K$ be its discriminant, let ${\mathcal T}_K$ be the relevant $p$-torsion group, and let

$${\mathcal R}_K:={\rm tor}_{\mathbb{Z}_p}\bigl({\rm \log}(U_K)/{\rm \log}(\overline E_K)\bigr)$$

be its normalized $p$-adic regulator. Let ${\rm \log}_\infty$ denote the complex logarithm. Regulator and torsion growth conjecture. There exists a constant $C_p>0$ such that, for every $K\in{\mathcal K}$,

$${\rm \log}_\infty(\# {\mathcal R}_K)\leq {\rm \log}_\infty(\# {\mathcal T}_K)\leq C_p\,{\rm \log}_\infty\bigl(\sqrt{|D_K|}\bigr).$$

Possibly, $C_p$ is independent of $p$. The source calls this a folk conjecture and attributes an earlier version to Gra77; it is motivated by extensive computations and is presented as relevant to questions including Leopoldt's conjecture.