Weak equivariant Beilinson–Burns–Flach conjecture

Let $A$ act on the motive $M$, and let $W$ be an $A$-submodule of $H^{i+1}_{\mathcal{M}/\mathcal{O}_K}(M,E(n))$. Let $r_{\mathrm{B}}$ be the Beilinson regulator, and suppose

$$r_{\mathrm{B}}(W)\otimes_{\mathbf{Q}}\mathbf{R}\cong H^{i+1}_{\mathcal{D}}(M,E_{\mathbf{R}}(n)).$$

Let $\vartheta_\infty(W)$ and $\vartheta'_\infty(W)$ be the elements of $K_0(A,\mathbf{R})$ arising from the two Deligne-cohomology exact sequences, and let $L^*$ be the leading term at $s=1-n$ of the dual equivariant $L$-function. Weak equivariant conjecture. There exists such a submodule $W$ for which

$$\hat{\delta}(L({}_A H^i(M),n))=\vartheta_\infty(W),\qquad \hat{\delta}(L^*)=\vartheta'_\infty(W).$$

This is a weakened form used because finite-dimensionality of the full motivic cohomology groups is generally out of reach; it retains the predicted equivariant special-value identities for a suitable submodule.