B-side Frobenius action correspondence for matrix factorizations

Let $(A,f)$ be a formal smooth commutative algebra of dimension $d<p$ over $\mathbf{k}$, with superpotential $f\in A$. Assume the HKR-type quasi-isomorphisms

$$(\bigwedge^*TA,\iota_{df})\simeq CC^*(\operatorname{MF}(A,f)),$$

and

$$(\Omega^*_{A}[[t]],td-df\wedge)\simeq CC^{S^1}_*(\operatorname{MF}(A,f)).$$

Under the identification $CC^{S^1}_*\oplus CC^{S^1}_*\theta\simeq CC^{\mathbb{Z}/p}_*$, these quasi-isomorphisms intertwine the $\mathbb{Z}/p$-equivariant cap product and a Frobenius $p$-linear graded multiplicative action on the de Rham side, where twisted functions $f\in A/\iota_{df}(TA)$ act by multiplication by $f^p$, and twisted vector fields $D\in\ker(\iota_{df}:TA\to A)/\iota_{df}(\bigwedge^2TA)$ act by

$$i^{[p]}_{D}:=(\iota_{D^p}-\mathcal{L}_{D}^{p-1}\iota_D)t^{\frac{p-1}{2}}.$$

This is a proposed B-side description of quantum Steenrod operations in arithmetic mirror symmetry. The correspondence is stated in the paper as expected, and its validity in the indicated generality remains open; related HKR results are known when $f$ has isolated singularities.