Tamagawa number conjecture for $(M,\mathcal{O}_F)$ in analytic rank zero

Let $M$ be a critical motive over $K$ with coefficient field $\mathcal{F}$, let $F=\mathcal{F}_\lambda$, and let $T$ be a stable $\mathcal{O}_F$-lattice in the associated representation $V$. Assume the Bloch–Kato Selmer groups $H^1_f(K,V)$ and $H^1_f(K,V^*(1))$ vanish. Choose an $\mathcal{F}$-basis $\gamma\in\bigwedge_\mathcal{F}^2H_B(M)^+$ whose comparison image is an $\mathcal{O}_F$-basis of $\bigwedge_{\mathcal{O}_F}^2T^*(1)$, and let $\delta^*$ be the dual basis in $\bigwedge_F^2D_{{\rm dR},\mathfrak{p}}(V)$. Tamagawa number conjecture. If $L(f,\chi^{-1},r)\ne0$, then $H^1_f(K,V)=H^1_f(K,V^*(1))=0$ and there is an $\mathcal{O}_F$-basis

$$\mathfrak{z}_\gamma\in\det_{\mathcal{O}_F}^{-1}(\mathbf{R}\Gamma(G_{K,S},T))$$

such that the localization and dual-exponential composition sends $\mathfrak{z}_\gamma$ to

$$\frac{L_S(f,\chi^{-1},r)}{\Omega_{\gamma,\delta}}\,\delta^*.$$

Here $L_S$ is obtained by removing the Euler factors at places in $S$. This is an integral refinement of the Tamagawa number conjecture in analytic rank zero, relating an integral determinant element to the normalized special value of the $L$-function. The source gives no resolution.