Co-Cartesian ramification-filtration diagrams for purely inseparable extensions

Let $K'/K$ be a geometric, purely inseparable, untwisted extension of geometric discrete valuation fields with finite exponent. Write $e$ for its ramification index, $f$ for its dual ramification index, and $\gamma:G_{K'}\to G_K$ for the induced isomorphism of absolute Galois groups. For $r\in\mathbb Q_{>1}$, consider the two commutative diagrams relating the graded ramification quotients and their character differentials as in Corollary. Co-Cartesian ramification-filtration conjecture. The two commutative diagrams in that corollary are co-Cartesian. In particular, if $K'/K$ has exponent $1$, then for every $r\in\mathbb Q_{>1}$ the sequence

$$G_K^{pr}/G_K^{pr+}\xrightarrow{\gamma^{-1}}G_{K'}^{er}/G_{K'}^{er+}\xrightarrow{\gamma}G_K^r/G_K^{r+}$$

is exact. This assertion concerns the compatibility of ramification filtrations under purely inseparable extensions; the supplied text gives no resolution status beyond stating it as a conjectural candidate.