Determinant de Rham cohomology formula for admissible connections

Let $f:X\to S$ be the family under consideration, let $D$ be the polar divisor, and write $K$ for the relevant function field. Let $E$ be a vector bundle on $X$ with an admissible connection

$$\nabla:E\to E\otimes\Omega^1_X\{{\mathcal D}\}.$$

The determinant de Rham cohomology is viewed as an element of $\Omega^1_K/d\log(K^\times)\otimes_{\mathbb Z}\mathbb Q$, and $\{c_1(\omega({\mathcal D})),\nabla\}$ is the class defined from the relative dualizing sheaf, the local connection matrices, and the transfer map described in the preceding construction.

Determinant de Rham formula.

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

This is presented as a summary of the results for admissible connections, with proofs stated to appear in the cited earlier work; the supplied text gives no resolution status beyond calling the statement a conjecture.