Compatibility of Ray–Singer metrics under the spectral-sequence identification

Let $(E,\mathbb{A})$ be a flat superconnection and let $(\mathcal{H},\tilde\nabla)$ be its Gauss–Manin cohomology bundle with induced flat superconnection. The associated spectral sequence gives an identification of determinant lines

$$\kappa:\det H(\mathcal{H},\tilde\nabla)\longrightarrow\det H(E,\mathbb{A}).$$

Compatibility conjecture. Under this identification of determinant lines, the Ray–Singer metric of the Gauss–Manin connection should map to the Ray–Singer metric of the original complex:

$$\|\cdot\|_{(\mathcal{H},\tilde\nabla)}\mapsto\kappa^*\|\cdot\|_{(E,\mathbb{A})}.$$

This asserts compatibility between the Ray–Singer metrics associated with a flat superconnection and its Gauss–Manin connection. The supplied text gives the proposed identity but no evidence of a proof or resolution.