The rationality conjecture for equivariant Artin L-function leading terms

Let $F'/F$ be the finite extension and let $G_{F'/F}$ be its Galois group. For the Tate motive $\mathbb{Q}(m)_{F'/F}$, let $L^*(\mathbb{Q}(m)_{F'/F})$ be its equivariant Artin $L$-function leading term, let $\lambda\in Z(\mathbb{Q}[G_{F'/F}])^{\times}$ be such that $\lambda L^*(\mathbb{Q}(m)_{F'/F})$ lies in the image of the reduced norm, and let $\partial$ be the connecting homomorphism in the localization sequence for $\mathbb{Q}[G_{F'/F}]\to\mathbb{R}[G_{F'/F}]$. Rationality conjecture. In $K_0(\mathbb{Q}[G_{F'/F}],\mathbb{R}[G_{F'/F}])$,

$$\partial\left(\operatorname{nrd}_{\mathbb{R}[G_{F'/F}]}^{-1}\left(\lambda L^*(\mathbb{Q}(m)_{F'/F})\right)\right)+[\Xi(\mathbb{Q}(m)_{F'/F}),\vartheta_{\infty}]=0.$$

This is identified with the corresponding formulation in the relative Grothendieck group via the canonical isomorphism. For negative $m$, the conjecture is stated to be equivalent to Gross's central conjecture, but its general status is not specified in the supplied text.