The asymptotic dimension conjecture for Hilbert cusp forms

Let $n=n(m)$, let $\Gamma_m\subset\mathrm{GL}^+_2(\mathbb{R})^n$ be a sequence of groups commensurable with a Hilbert modular group, and suppose that whenever $m\ne m'$ and $n=n(m)=n(m')$, the groups $\Gamma_m$ and $\Gamma_{m'}$ are not conjugate in $\mathrm{GL}^+_2(\mathbb{R})^n$. Let $\mathbb{H}^n$ be the product of $n$ upper half-planes, let $N$ denote the norm defined in the paper, and let $S_k(\Gamma_m)$ denote the space of cusp forms of weight $k$. Asymptotic dimension conjecture. One has

$$\lim_{m\rightarrow\infty}(4\pi)^{-n}\operatorname{vol}(\Gamma_m\backslash\mathbb{H}^n)=\infty,$$

and

$$\lim_{|k|+m\rightarrow\infty}\frac{\dim S_k(\Gamma_m)-(4\pi)^{-n}N(k-\mathbf{1})\operatorname{vol}(\Gamma_m\backslash\mathbb{H}^n)}{(4\pi)^{-n}N(k-\mathbf{1})\operatorname{vol}(\Gamma_m\backslash\mathbb{H}^n)}=0.$$

The conjecture proposes a volume-growth condition and a uniform asymptotic formula for dimensions of Hilbert cusp-form spaces; the paper states that it proves the volume part, while the full dimension asymptotic remains conjectural.