Nevanlinna's generalized volume-capacity conjecture

Let $h$ be a continuous, increasing, concave function defined and positive in some right neighborhood of $0$. For a set $E\subset\mathbb{R}$, define its $h$-volume by

$$m_h(E):=\lim_{\varepsilon\to 0+}\inf_{\{(x_j,r_j)_{j\in\mathbb{N}}\}\in\mathcal{I}(E,\varepsilon)}\sum_j h(r_j),$$

where

$$\mathcal{I}(E,\varepsilon):=\left\{(x_j,r_j)_{j\in\mathbb{N}}\mid\bigcup_j U_{r_j}(x_j)\supset E,\quad \forall j\quad r_j<\varepsilon\right\}.$$

Assume that

$$\int_0^{\bullet}\frac{h(t)}{t}\,dt$$

diverges and that $E$ is a closed set with finite $h$-volume. Nevanlinna's conjecture. Then $\operatorname{Cap}(E)=0$. This conjecture generalizes the Erdős–Gillis theorem for the logarithmic measuring function $h(t)=1/|\log t|$, which gives zero capacity when the corresponding $h$-volume is finite. The source mentions that the conjecture goes back to Nevanlinna; its resolution is not specified here.