Productivity of paracompactness and closed images of real GO-spaces

One of the most important problems concerning paracompactness is the characterization of productively paracompact spaces, i.e., the spaces whose product with every paracompact space is paracompact. To this end, an infinite topological game introduced by Telgársky is very useful. Telgársky proved that if X is a paracompact space and the first player has a winning strategy in his game played on the space X, then the space X is productively paracompact. In 2009, Alster conjectured that a paracompact space X is productively paracompact if and only if the first player has a winning strategy in Telgársky's game played on the space X. We prove that Telgársky's conjecture is true in the classes of closed images of real generalized ordered spaces, subspaces of the Sorgenfrey line, and the Michael line. In particular we show that a space that is a closed image of an arbitrary subspace of the Sorgenfrey line is productively paracompact if it is countable. We also show that every separable, productively paracompact space has the Hurewicz covering property.