The $p$-adic equivariant Tamagawa number conjecture for \mathbb{G}_m

Let $L/K$ be the extension and $G={\rm Gal}(L/K)$, let $C=C_{L,S,T}$ be the perfect complex over $\mathbb{Z}[G]$, and for a prime $p$ let $C_p$ be its $p$-adic realization. For an isomorphism of fields $j:\mathbb{C}\cong\mathbb{C}_p$, let $z_p=z^j_{L/K,S,T}$ be the associated zeta element. The $p$-adic equivariant Tamagawa number conjecture. For every such $j$, one has

$${\rm Nrd}_{\mathbb{Q}_p[G]}(K_1(\mathbb{Z}_p[G]))\cdot z_p={\rm d}_{\mathbb{Z}_p[G]}(C_p)^{\rm pb}.$$

The source states that this equality for all primes $p$ is equivalent to the equivariant Tamagawa number conjecture for $\mathbb{G}_m$ relative to $L/K$; no resolution status is supplied.