Higher-rank Gauß–Manin determinant formula for irregular connections on curves

Let $C$ be a curve over a function field $K$ of an algebraically closed field $k$ of characteristic $0$, let $U\subset C$ be an open set with $D=X-U=\sum_i c_i$ consisting of $K$-rational points, and let $({\mathcal E},\nabla)$ be a rank-$r$ connection on $U$ with vertical curvature. Let $m_i$ be the multiplicity of the relative connection at $c_i$, set ${\mathcal D}=\sum_i m_i c_i$ and ${\mathcal D}'={\mathcal D}-D$, and assume

$$\nabla:E\longrightarrow E\otimes\Omega^1_C\langle D\rangle({\mathcal D}')$$

and that the image of each local matrix $g_i$ is invertible in $GL(r,\mathcal O_{m_i c_i})$.

Higher-rank Gauß–Manin determinant formula. Assuming these conditions, one has

$$\operatorname{det}H^*_{DR}(U,({\mathcal E},\nabla))^{-1}=\{c_1(\omega({\mathcal D})),\nabla\}\in(\Omega^1_K/d\log K^*)\otimes_{\mathbb Z}\mathbb Q.$$

This is a proposed higher-rank extension of the determinant formula for de Rham cohomology of irregular connections on curves. The supplied text gives the formula under the stated assumptions but provides no resolution status, so its conjectural status remains unclear.