Looijenga–Saper–Stern conjecture for modularity in L2-cohomology

Let $X$ be a Shimura variety with special cycles $\mathcal Z(m)$ whose generating series is modular in cohomology, and let $X^*$ be its Satake–Baily–Borel compactification. Looijenga–Saper–Stern conjecture. The classes $[\mathcal Z(m)]$ are defined in the $L^2$-cohomology of $X$, and their generating series is a modular form. The conjecture seeks an $L^2$-cohomological realization of the special-cycle classes compatible with modularity; the source presents it as a natural conjectural extension of known cohomological modularity results.