Modularity conjecture for higher Heegner cycles

Let $X=X_0(N)$, let $\mathcal{Y}$ be the Kuga–Sato variety, and let

$$\phi_\kappa=\sum_{m>0,\,\mu}Z_\kappa(m,\mu)q^me_\mu.$$

Let $G\in S_{2\kappa}(\Gamma_0(N))$ be a normalized newform, let $g\in S_{1/2+\kappa,\rho_L}$ be the newform corresponding to $G$ under the Shimura correspondence, and let $f\in H_{3/2-\kappa,\bar\rho_L}$ satisfy

$$\xi_{3/2-\kappa}(f)=\frac{1}{\langle g,g\rangle_{\operatorname{Pet}}}g.$$

Modularity conjecture. The function

$$\phi_\kappa\in \operatorname{CH}^{\kappa}(\mathcal{Y})\otimes S_{1/2+\kappa,\rho_L}$$

is a cusp form, where $\operatorname{CH}^{\kappa}(\mathcal{Y})$ is the codimension-$\kappa$ Chow group, and its $G$-component is

$$\phi_\kappa^G=g(\tau)\otimes Z_\kappa(f).$$

For $\kappa=1$, modularity of Heegner divisors is known by work of Gross, Kohnen and Zagier, Borcherds, and Bruinier and Ono. The corresponding arithmetic generating function is also expected to be modular, while the stated Chow-valued conjecture is established after replacing the Chow group by the algebraic Heegner-cycle quotient.