Mochizuki's absolute anabelian conjecture without a fixed Galois-group isomorphism

Let $K$ and $L$ be finite extensions of $\mathbb{Q}_p$, and let $X/L$ and $Y/K$ be hyperbolic curves. Write $\operatorname{Isom}(\operatorname{\pi_1^{alg}}(X),\operatorname{\pi_1^{alg}}(Y))/\sim$ for the isomorphisms of algebraic étale fundamental groups modulo composition with inner automorphisms. Mochizuki's absolute anabelian conjecture. The map

$$\operatorname{Isom}(X,Y)\to \operatorname{Isom}(\operatorname{\pi_1^{alg}}(X),\operatorname{\pi_1^{alg}}(Y))/\sim$$

is bijective. This strengthens the preceding anabelian statement by allowing the two curves to lie over possibly different $p$-adic fields, and the source presents it as a conjecture prompted by Mochizuki's question.