Logarithmic Kummer pro-étale Poincaré sequences for differential crystals

Let $X$ be a smooth space in the setting of the paper, and consider the projective maps

$$g:X_{\text{Kummer-pro-étale}}\rightarrow X_{\text{Kummer-ét}},\qquad f:X_{\text{Kummer-pro-étale}}\rightarrow X.$$

Let $M$ be a corresponding $Z$-projective differential crystal spectrum. Logarithmic crystal Poincaré conjecture. The logarithmic versions of the strictly exact Poincaré long exact sequences for $M$ should hold, as in the pseudo-rigid analytic situation.

This is explicitly presented as a conjecture based on the preceding Kummer pro-étale extension claim. The supplied text gives no proof or resolution.