The rank-two Euler-system conjecture for Beilinson–Flach classes

Let $T$ be the rank-four representation, let $\mathcal{N}(\mathcal{P})$ be the specified set of auxiliary integers, and let $\mathcal{L}_{m,V}$ be the equivariant Perrin–Riou regulator. Choose an eigenvector basis \\{v_{\lambda\mu}\} of $\mathbb{D}_{\mathrm{cris}}(V)$ and its dual basis $\\{v_{\lambda\mu}^*\\}$, with $\lambda,\mu\in\\{\alpha,\beta\\}$. Rank-two Euler-system conjecture. There exists a collection of classes

$$\textup{BF}_m\in\bigwedge^2H^1_{\mathrm{Iw}}(\mathbb{Q}(\mu_m),T),$$

for all $m\in\mathcal{N}(\mathcal{P})$, forming a rank-two Euler system, such that

$$\left\langle\mathcal{L}_{m,V}(\textup{BF}_m),v_{\lambda,\mu}^*\right\rangle=\textup{BF}_{\lambda,\mu,m}$$

for every $\lambda,\mu\in\\{\alpha,\beta\\}$. This would package the four $p$-stabilised Beilinson–Flach classes into one rank-two Euler system.