Conjectural Chern-class compatibility for toroidal embeddings

Let $L\subset\mathbb{C}$ be a number field with ring of integers $\mathcal{O}_L$, let $X/L$ be a smooth projective variety equipped with a toroidal embedding $Z\subset X$, and let $\mathcal{V}$ be a rank $n$ vector bundle on $X$ whose restriction to each irreducible component of $Z$ is trivial. Let $\rho:\pi_1(Z(\mathbb{C}))\to\operatorname{GL}_n(\mathcal{O}_L)$ be the associated local system, let $s:Z(\mathbb{R}_{\geq 0})\to Z(\mathbb{C})$ be the toroidal map, and let $b_i$ be the Borel and Soulé regulator class, with pullback $f_\rho^*(b_i)\in\operatorname{Ext}^1_{\mathcal{C}}(1_{\mathcal{C}},1_{\mathcal{C}}(i))\otimes H^{2i-1}(Z(\mathbb{R}_{\geq 0}),\mathbb{Z})$. Conjectural Chern-class compatibility. For every $i$, one has

$$f_\rho^*(b_i)=s^*(c_i(\mathcal{V})|_Z).$$

This conjecture proposes an étale analogue of Looijenga's compatibility between regulator classes and Chern classes, extending the stated theorem for line bundles. In the rank-one case with $i=1$, it identifies the pulled-back first Chern class with the unit $\alpha$ defining the corresponding monodromy representation.