Burns–Flach conjecture at left-of-center special values

Let $A$ be the semisimple coefficient algebra acting on the motive $M$, let $d$ be its relevant dimension, let $i$ be an integer, and let $n>\frac{i}{2}+1$ be an integer. For an embedding $\sigma:E\hookrightarrow\mathbf{C}$, write $L({}_{A^{\operatorname{op}}}H^{2d-i}(M^*),s)^\sigma$ for the corresponding component, and let $\operatorname{rr}_{A,\sigma}$ be the reduced-rank morphism. Let $L^*\in Z(A_{\mathbf{R}})^\times$ be the componentwise leading Taylor coefficient at $s=1-n$, and let $\vartheta'_\infty(M,i,n)\in K_0(A,\mathbf{R})$ be the canonical element obtained from the alternative Deligne-cohomology exact sequence. Burns–Flach's conjecture. For every embedding $\sigma:E\hookrightarrow\mathbf{C}$,

$$\operatorname{ord}_{s=1-n}L({}_{A^{\operatorname{op}}}H^{2d-i}(M^*),s)^\sigma=\operatorname{rr}_{A,\sigma}\bigl(H^{i+1}_{\mathcal{M}/\mathcal{O}_K}(M,E(n))\bigr),$$

and

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

This reformulates the equivariant special-value conjecture at integers to the left of the central point, assuming meromorphic continuation of the equivariant $L$-function. The source gives no resolution.