Arithmetic modularity conjecture for higher Heegner cycles

Let $\widehat{\operatorname{CH}}^{\kappa}(\mathcal{Y})$ be the arithmetic Chow group and define the arithmetic generating function

$$\widehat{\phi}_\kappa=\sum_{m>0,\,\mu}\widehat{\mathcal{Z}}_\kappa(m,\mu)q^me_\mu\in\widehat{\operatorname{CH}}^{\kappa}(\mathcal{Y})\otimes\mathbb{C}[[q]].$$

Arithmetic modularity conjecture. The function $\widehat{\phi}_\kappa$ is a cusp form. For $\kappa=1$, the source states that arithmetic modularity on the modular curve $X_0(N)$ has been proved.