Kudla's arithmetic modularity conjecture

Let $G$ be the relevant reductive group, let $X_K$ be the associated Shimura variety, and let $n$ be a nonnegative integer. For a Schwartz function $\varphi$ in the finite adelic Schwartz space, write $Z(g,\varphi)_K$ for the formal generating function and let ${\mathscr A}_{m/2,\chi}(G(\mathbb A))$ be the indicated space of automorphic forms. The conjecture concerns the generating function

Kudla's arithmetic modularity conjecture. The formal generating function $Z(g,\varphi)_K$ converges absolutely and defines an element in

$${\mathscr A}_{m/2,\chi}(G(\mathbb A)) \otimes {\mathrm{CH}}^{n}(X_K)_\mathbb C.$$

This is the arithmetic analogue of the modularity of classical and geometric theta functions: it predicts that the generating series of arithmetic cycles is an automorphic form with values in the complexified Chow group. The supplied text does not state a resolution, so the conjecture is recorded as open.