The comeagre-conjugacy conjecture for countable index in Polish groups

Let $G$ be a Polish group, and let $C\subseteq G$ be a comeagre conjugacy class, meaning a conjugacy class whose complement is meagre. Say that $G$ has the countable index property when every subgroup of index at most countable is open. Comeagre-conjugacy conjecture. Every Polish group with a comeagre conjugacy class has the countable index property. The source motivates this claim by proving that such a group has the normal countable index property, but gives no resolution of the stronger assertion.