Compatibility map to Z/pα without assuming an E1-algebra map S_{R∞}→S/pα

Develop a construction of the canonical comparison map from the q-Hodge filtration on the derived q-de Rham complex _{R/A} to the q-Hodge filtration on _{(Z/p^α)/Z_p} for A-algebras R satisfying the hypotheses of Theorem 2.8(2) (existence of an E1-lift S_{R∞}), without additionally assuming the existence of an E1-algebra map S_{R∞}→S/p^α.

Background

In Section 3.2 the authors define a q-Hodge filtration on {(Z/p^α)/Z_p} by base change and show compatibility in several cases. They then point out a limitation: their current method requires an additional E1-map from the lift S{R∞} to S/p^α.

Removing this auxiliary assumption would broaden the applicability of the compatibility construction between q-Hodge filtrations.