Some remarks on a Telgarsky's conjecture concerning products of paracompact spaces

R. Telgarsky conjectured that if X is a paracompact space then the product X×Y is paracompact for every paracompact space Y if and only if the first player of the G(DC,X) game, introduced by R. Telgarsky, see [R. Telgarsky, Spaces defined by topological games, Fund. Math. 88 (1975) 193–223], has a winning strategy. The paper contains some results supporting this conjecture.