Atsuta–Kataoka's shifted Fitting ideal conjecture for class groups

Let $R=\mathbf{Z}[1/2][G]^-$, let $\operatorname{Cl}^T(H)^-$ be the minus part of the $T$-smoothed class group, and for each ramified place $v$ let $A_v=\mathbf{Z}[G/I_v]/(g_v)$. With $h_v^-$ and $\operatorname{Fitt}_R^{[1]}(A_v^-)$ defined as in the source, and with $\Theta_{S_\infty,T}$ the smoothed Stickelberger element, Atsuta–Kataoka's conjecture. One should have

$$\operatorname{Fitt}_R(\operatorname{Cl}^T(H)^-)=\left(\prod_{w\in S_{\mathrm{ram},H}}h_v^-\operatorname{Fitt}_R^{[1]}(A_v^-)\right)\Theta_{S_\infty,T}.$$

The conjecture is motivated by shifted Fitting ideals and gives a formula for the Fitting ideal of the class group itself. The source states that the equivariant Tamagawa number conjecture implies it, but does not otherwise resolve it.