Explicit reciprocity law for the GSp4 times GL2 Euler-system class

Let $\underline{\pi}$ and $\underline{\sigma}$ be the Coleman families and let $\mathbf{z}_1(\underline{\pi}\times\underline{\sigma},\gamma_S)$ be the associated Euler-system cohomology class. Let $\mathcal{L}^{\operatorname{PR}}$ be the Perrin--Riou big logarithm, let $\operatorname{ES}_{\pi_P}^1$ and $\operatorname{ES}_{\sigma_Q}^1$ be the Eichler--Shimura isomorphisms, and let $\mathcal{L}_{p,\gamma_S}^{\operatorname{imp}}(\underline{\pi}\times\underline{\sigma})$ be the improved $p$-adic $L$-function. Explicit reciprocity conjecture. Under the running assumptions, for all $(P,Q)$ in the geometric range,

$$\left\langle \mathcal{L}^{\operatorname{PR}}\bigl(\mathbf{z}_1(\underline{\pi}\times\underline{\sigma},\gamma_S)\bigr)(P,Q),\operatorname{ES}_{\pi_P}^1(\xi_P)\otimes\operatorname{ES}_{\sigma_Q}^1(\eta_Q)\right\rangle=\mathcal{L}_{p,\gamma_S}^{\operatorname{imp}}(\underline{\pi}\times\underline{\sigma})(P,Q).$$

This predicts that the Perrin--Riou logarithm of the Euler-system class recovers the improved $p$-adic $L$-function; no resolution is supplied in the source.