Fukaya–Kato equivariant local Tamagawa-number conjecture

Let $\Lambda$ be the Iwasawa algebra of the relevant $p$-adic Lie group, let $\widetilde\Lambda$ be its unramified coefficient extension, and let $\mathbb T$ be the associated big Galois representation. For every finite coefficient representation $\rho:G\to\operatorname{GL}_n(\mathcal O)$, write $T(\rho^*)$ for the specialized lattice. Equivariant local Tamagawa-number conjecture. There exists a unique isomorphism

$$\epsilon_{p,\Lambda}(\mathbb T):\mathbf{1}_{\widetilde\Lambda}\to\bigl(\mathcal D_\Lambda(\mathrm{R}\Gamma(\mathbb Q_p,\mathbb T))\mathcal D_\Lambda(\mathbb T)\bigr)_{\widetilde\Lambda}$$

such that

$$\mathcal O^n\otimes_\Lambda\epsilon_{p,\Lambda}(\mathbb T)=\epsilon_{p,\mathcal O}(T(\rho^*));$$

its assignment is required to satisfy the compatibility conditions described in the source. The source states that the commutative-$\Lambda$ case was proved by S. Yasuda, while the noncommutative case was expected to follow by extending those methods.