Generalized Gross--Kudla conjecture for Chow cycles

Let $r$ be a positive integer and let $(\kappa,\lambda)\in\mathcal A_r^{(2)}$. Write $X_r=X_1(Np^r)$, $f=\mathbf f_\kappa$, and $g=\mathbf g_\lambda$. Assume that $f$ and $g$ are newforms and that $s_\kappa=r=s_\lambda$. Define

$$\Omega_{f\otimes {\rm ad}(g)}:=a_p(f)^{-2r}\,\langle f,f\rangle\,\langle g,g\rangle^2.$$

Let $\Delta_r^{\circ}(\kappa,\lambda)\in {\rm CH}_0^2(X_r^3;\mathbb{Q}(\mu_{p^r}))\otimes_{\mathbb{Z}}F_{\kappa,\lambda}$ be the specified cycle, and let $\langle\,\ \rangle_{\mathrm{BB}}$ be the Beilinson--Bloch height pairing. Generalized Gross--Kudla conjecture.

$$\left\langle \Delta_r^{\circ}(\kappa,\lambda),\Delta_r^{\circ}(\kappa,\lambda)\right\rangle_{\mathrm{BB}}=\frac{p^{-r}\cdot C(\kappa,\lambda)}{\Omega_{f\otimes {\rm ad}(g)}}\Lambda'(f\otimes g\otimes g^c,\psi_\kappa^{-\frac12},2),$$

where $C(\kappa,\lambda)$ is generically a non-zero algebraic constant interpolating $p$-adically as $(\kappa,\lambda)$ varies. This conjecture extends the relation between heights of Chow cycles and central derivatives of triple product $L$-functions; the source states that it is in line with results of Yuan--Zhang--Zhang, which partially prove the corresponding Gross--Kudla conjecture.