Conjecture that Arakelov and theta densities coincide for homogeneous subsets

Let $\mathcal{X}$ be a regular projective arithmetic variety of dimension $n$, and let $\overline{\mathcal{L}}$ be an ample line bundle on $\mathcal{X}$. Let

$$\mathcal{E}\subset \bigcup_{d>0}\operatorname{H}^0(\mathcal{X},\overline{\mathcal{L}}^{\otimes d})$$

be a subset such that whenever $\sigma\in\mathcal{E}$ and $k\in\mathbb{Z}$, one has $k\sigma\in\mathcal{E}$. Arakelov–theta density conjecture. Then

$$\mu_{\operatorname{Ar}}(\mathcal{E})=\mu_{\theta}(\mathcal{E}).$$

The conjecture proposes that the Arakelov density and the $\theta$-density agree for every subset of sections stable under multiplication by integers, extending the coincidence already observed for subsets defined by reduction modulo a fixed integer. Its general validity is not established in the supplied text.