Selmer criterion for plectic Heegner classes

Let $A$ be the abelian variety, $E$ the number field, $L_{\mathfrak c}$ the anticyclotomic extension, and let $\chi\colon\mathcal{G}_{\mathfrak c}\to\overline{\mathbb{Q}}^\times$ be a primitive character. Let $r$ be the number of primes in $\mathcal{S}$, let $\kappa^{\chi}_{A,\mathcal{S}}$ be the plectic Heegner class, and write $r_\mathrm{an}(A/E,\chi)$ for the analytic rank. The class is called Selmer when

$$\kappa^{\chi}_{A,\mathcal{S}}\in \mathrm{H}^1_f(E_{\otimes,\mathcal{S}},V_p(A))\otimes \mathrm{H}^1_f(L_{\mathfrak c},V_p(A))^\chi.$$

Selmer criterion for plectic Heegner classes. If $\chi$ is primitive, then

$$\kappa^{\chi}_{A,\mathcal{S}}\text{ is Selmer}\quad\iff\quad r_\mathrm{an}(A/E,\chi)\ge r.$$

In particular, if $r_\mathrm{an}(A/E,\chi)<r$, then $\kappa^{\chi}_{A,\mathcal{S}}\ne 0$. This extends conjectures on plectic and mock plectic classes and relates the Selmer property of the constructed class to the analytic rank.