The signed decomposition conjecture for Beilinson–Flach classes

Let $\mathcal{N}(\mathcal{P})$ be the specified set of auxiliary integers, let $\mathbb{T}=T\otimes\Lambda^\iota$, and let $M_{\log}$ and $Q$ be the matrices appearing in the signed Coleman-map decomposition. For each $m\in\mathcal{N}(\mathcal{P})$ and each \bullet,\circ\in\\{\\#,\flat\\}, consider the signed Selmer group $H^1_{\mathcal{F}_{\bullet,\circ}}(\mathbb{Q}(\mu_m),\mathbb{T})$. Signed decomposition conjecture. There exists a non-zero $r_0\in\mathbb{Z}$ and classes

$$\textup{BF}_{\bullet,\circ,m}\in H^1_{\mathcal{F}_{\bullet,\circ}}(\mathbb{Q}(\mu_m),\mathbb{T})$$

for which

r_0\begin{pmatrix}\textup{BF}_{\alpha,\alpha,m}\\\\textup{BF}_{\alpha,\beta,m}\\\\textup{BF}_{\beta,\alpha,m}\\\\textup{BF}_{\beta,\beta,m}\end{pmatrix}=Q^{-1}M_{\log}\begin{pmatrix}\textup{BF}_{\\#,\\#,m}\\\\textup{BF}_{\\#,\flat,m}\\\\textup{BF}_{\flat,\\#,m}\\\\textup{BF}_{\flat,\flat,m}\end{pmatrix}.

This is a weaker alternative to the rank-two Euler-system conjecture and is intended to make Euler-system methods available in cases where it can be verified.