The countable-index conjecture for automatic continuity in Polish groups

Let $G$ be a Polish group. Say that $G$ has the countable index property when every subgroup of index at most countable is open, and say that $G$ has the automatic continuity property when every homomorphism from $G$ into a separable topological group is continuous. Countable-index conjecture. Every Polish group with the countable index property also has the automatic continuity property. The conjecture asks whether the countable index and automatic continuity properties coincide for Polish groups; the source gives no resolution.