Modularity conjecture for the $K3$ hypersurfaces in the paper

Consider each $K3$ hypersurface listed above, after desingularization when necessary. $K3$ modularity conjecture. Each such $K3$ hypersurface is modular with the weight-$3$ newform given in the column containing $\operatorname{Re}(\widetilde{m}(\tau))$. This extends the modularity phenomenon from singular $K3$ surfaces to the listed families; the source gives no resolution of the assertion.