Epsilon-class formula for determinant de Rham cohomology

Let $X\to S$ be the family under consideration, let ${\mathcal D}$ be the fixed divisor, and let $(E,\nabla,\mathcal L,\mu)$ consist of an admissible absolute connection, a line bundle $\mathcal L$ on $X$, and an isomorphism

$$\mu:E_{\mathcal D}\xrightarrow{\sim}E_{\mathcal D}\otimes\mathcal L_{\mathcal D}$$

compatible with the induced connection along ${\mathcal D}$. The construction associates to such a tuple a class $\epsilon(E,\nabla,\mathcal L,\mu)\in AD^2(X)$; in the distinguished case $\mathcal L=\omega({\mathcal D})$ and $\mu=\nabla_{X/S,\mathcal D}$, write it as $\epsilon(E,\nabla)$.

Epsilon-class conjecture. If $\nabla$ is an integrable, admissible, absolute connection, then

$$\det H^*_{DR/S}(E,\nabla)=-f_*\epsilon(E,\nabla).$$

This is stated as a restatement of the paper's main conjecture. The class $\epsilon(E,\nabla)$ was constructed immediately beforehand and its pushforward was identified with the previously defined expression; the supplied text gives no evidence that the conjecture has been resolved.