Automatic continuity criterion for epimorphisms onto discrete groups

Let $G$ be a group such that every torsion subgroup of $G$ is finite, $G$ contains neither $\mathbb{Q}$ nor the $p$-adic integers $\mathbb{Z}_p$ as a subgroup for any prime $p$. Let $L$ be a locally compact Hausdorff group and let $L\twoheadrightarrow G$ be an epimorphism.

Automatic continuity conjecture. Every epimorphism from a locally compact Hausdorff group $L$ to $G$ is continuous if and only if $G$ has no non-trivial finite normal subgroups.

This conjecture would complete the known sufficient criterion for automatic continuity of epimorphisms onto discrete groups: under the stated torsion and subgroup hypotheses, the absence of non-trivial finite normal subgroups implies continuity. The converse follows from the existence of discontinuous epimorphisms onto groups with suitable finite normal subgroups, while the general equivalence is posed here as a conjecture.