The twisted-diagonal-cycle factorisation conjecture for the $a$-region

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ with root number $-1$ and no endoscopic specialisations, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ family, and let $\operatorname{Log}_c$ be the Perrin-Riou exponential map. Let $\mathrm{LF}_{\underline{\Pi}}^{\dagger}$ be the Lemma–Flach family and let $\Delta_{\underline{\Pi}\otimes\operatorname{ad}\,\underline{\sigma}}^{\dagger}$ be the conjectural family of twisted diagonal cycles in the stated Selmer group. Twisted-diagonal-cycle factorisation conjecture. There is a non-zero $\mathcal C_a\in\mathcal R_{\underline{\Pi}}[1/p]$ such that

$$L_p^{(a)}(\underline{\Pi}\times\operatorname{ad}\,\underline{\sigma})^2(P,Q,Q)=S(M,\underline{\Pi})\mathcal C_a(P)\operatorname{Log}_c(\Delta_{\underline{\Pi}\otimes\operatorname{ad}\,\underline{\sigma}}^{\dagger})^2\operatorname{Log}_c(\mathrm{LF}_{\underline{\Pi}}^{\dagger}).$$

Here $\mathcal C_a$ interpolates explicit algebraic fudge factors and $S(M,\underline{\Pi})$ is the stated $p$-adic period. This conjecture combines the two wall-crossing contributions: a twisted diagonal cycle and a Lemma–Flach class.