  The classical additive Deligne-Simpson problem is the existence problem for
Fuchsian connections with residues at the singular points in specified adjoint
orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem
in terms of quiver varieties. A more general version of this problem, solved by
Hiroe, allows additional unramified irregular singularities. We apply the
theory of fundamental and regular strata due to Bremer and Sage to formulate a
version of the Deligne-Simpson problem in which certain ramified singularities
are allowed. These allowed singular points are called toral singularities; they
are singularities whose leading term with respect to a lattice chain filtration
is regular semisimple. We solve this problem in the important special case of
connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and
possibly an additional regular singular point at infinity. We also give a
complete characterization of all such connections which are rigid, under the
additional hypothesis of unipotent monodromy at infinity.
