Limit-height conjecture for translated complete intersections in toric varieties

Let $\mathbb{K}$ be a number field with set of places $\mathfrak{M}$, let $\mathbb{T}$ be a split algebraic torus over $\mathbb{K}$ of dimension $n$ with character lattice $M$, and let $k\leq n$. Let $\bm{f}=(f_1,\ldots,f_k)$ be a family of nonzero Laurent polynomials in $\mathbb{K}[M]$. Let $X$ be a complete toric variety with torus $\mathbb{T}$, equipped with semipositive toric metrized divisors $\overline{D}_0,\ldots,\overline{D}_{n-k}$. For a family of points $\bm{t}=(t_1,\ldots,t_k)$, write $\bm{t}^*\bm{f}=(t_1^*f_1,\ldots,t_k^*f_k)$ and let $Z_{\mathbb{T}}(\bm{t}^*\bm{f})$ denote the cycle obtained by properly intersecting the corresponding twisted divisors on $\mathbb{T}_{\overline{\mathbb{K}}}$. Limit-height conjecture. For each sequence $(\bm{\omega}_\ell)_\ell$ of torsion points in $\mathbb{T}(\overline{\mathbb{K}})^k$ whose projection to $(\mathbb{T}^k/\mathbb{T})(\overline{\mathbb{K}})$ via the diagonal action is strict, one has

$$\lim_{\ell\to\infty}\operatorname{h}_{\overline{D}_0,\ldots,\overline{D}_{n-k}}(Z_{\mathbb{T}}(\bm{\omega}_\ell^*\bm{f})) =\sum_{v\in\mathfrak{M}}n_v\operatorname{MI}_M(\vartheta_{\overline{D}_0,v},\ldots,\vartheta_{\overline{D}_{n-k},v},\rho_{f_1,v}^{\vee},\ldots,\rho_{f_k,v}^{\vee}).$$

Here $n_v$ is the weight of $v$, $\vartheta_{\overline{D}_i,v}$ is the $v$-adic roof function of $\overline{D}_i$, and $\rho_{f_j,v}^{\vee}$ is the Legendre-Fenchel dual of the $v$-adic Ronkin function of $f_j$. This conjectural formula would express the limiting height of translated higher-codimensional complete-intersection cycles using only toric metrized-divisor data and the arithmetic Ronkin data of the defining Laurent polynomials; it strengthens earlier guesses of Gualdi and Sombra and remains unresolved here.