Is each Choquet topological group strong Choquet?

A topological space $$X$$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.

It is known that a metrizable space $$X$$ is

$$\bullet$$ Choquet if and only if it contains a dense complete-metrizable subspace;

$$\bullet$$ strong Choquet if and only if it is complete-metrizable.

Since a topological group is complete-metrizable if and only if it contains a dense complete-metrizable subspace, we obtain the following characterization.

Theorem. A metrizable topological group is Choquet if and only if it is strong Choquet.

Corollary. A cosmic topological group is Choquet if and only if it is strong Choquet.

Let us recall that a regular topological space is cosmic if it has a countable network of the topology. It is easy to show that a topological group is second-countable if it is cosmic and Baire. That is why Corollary follows from Theorem.

Theorem and Corollary suggest the following intriguing

Problem. Is each Choquet topological group strong Choquet?