Excision for periodic dual Hopf cyclic homology of comodule algebras

Let $B$ be a Hopf algebra with an invertible antipode and let $X$ be a stable anti-Yetter–Drinfeld module. Let

$$0\longrightarrow J\longrightarrow A\longrightarrow A/J\longrightarrow 0$$

be a sequence of $B$-comodule algebras such that both $J$ and $A$ are $B$-co-projective. Excision conjecture. One has a 6-term exact sequence in periodic cyclic homology:

$$\begin{CD} HP_1(J,B;X) @>>> HP_1(A,B;X) @>>> HP_1(A/J,B;X)\\ @AAA @. @VVV\\ HP_0(A/J,B;X) @<<< HP_0(A,B;X) @<<< HP_0(J,B;X) \end{CD}$$

This is the periodic analogue of excision for dual Hopf cyclic homology. The surrounding text states a homotopy cofibration theorem for the corresponding mixed complexes and notes that the usual non-equivariant result is Wodzicki's excision theorem, but gives no explicit resolution status for this periodic formulation.