Perrin-Riou's equivariant epsilon-isomorphism conjecture

Let $K$ be a finite extension of $\mathbb{Q}_p$, let $G_K$ be its absolute Galois group, let $V=\mathbf{Q}_p\otimes_{\mathbf{Z}_p}T$ be a potentially semistable representation of $G_K$ with $T$ a $\mathbf{Z}_p$-lattice, and let $L/K$ be a finite abelian extension. Write $G=\operatorname{Gal}(L/K)$, let $\Delta_{\mathrm{EP}}(L/K,T)$ be the canonical $\mathbf{Z}_p[G]$-lattice in the Euler–Poincaré line, and let $\delta_{V,L/K}:\Delta_{\mathrm{EP}}(L/K,V)\xrightarrow{\sim}\mathbf{Q}_p[G]_{V,L/K}$ be the canonical trivialization.

$C_{\mathrm{EP}}(L/K,V)$. The map $\delta_{V,L/K}$ sends $\Delta_{\mathrm{EP}}(L/K,T)$ onto $\mathbf{Z}_p[G]_{V,L/K}$.

This is an integral refinement of the canonical trivialization of the Euler–Poincaré line for potentially semistable representations. The source presents it as a conjecture and gives references to work of Fukaya–Kato, Perrin-Riou and Kato; no resolution is stated here.