Chen's conjecture on the essential minimum and asymptotic maximal slope

Let $X$ be a projective variety over a global field, let $D$ be a divisor on $X$, and let $\overline{D}$ be a metrized divisor. For each integer $n\geq 1$, set $V_n=H^0(X,nD)$ and equip it with the supremum norms at the places of the global field. The essential minimum is denoted by $\operatorname{\zeta_{\mathrm{ess}}}(\overline{D})$, and, when $D$ is big, the limit of the normalized maximal slopes is denoted by $\operatorname{\widehat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\overline{D})$. Chen's conjecture. If $\overline{D}$ is semi-positive and $D$ is big, then

$$\operatorname{\zeta_{\mathrm{ess}}}(\overline{D})=\operatorname{\widehat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\overline{D}).$$

This conjecture relates the essential minimum, an arithmetic-geometric invariant of the metrized divisor, to the asymptotic maximal slope of its spaces of global sections. The statement remains open in general, particularly beyond the toric setting.