The TWO strategy conjecture for the countable-dimensional game

Let $X$ be a $T_1$ space, and let $G_c(\mathcal O,\mathcal O)$ denote the compact-open selection game in which ONE and TWO choose open covers and TWO selects finite subfamilies. Write $TWO\uparrow G_c(\mathcal O,\mathcal O)$ for the assertion that TWO has a winning strategy in this game. Corollary 4.2 characterizes $\sigma$-zero-dimensionality together with related winning conditions. The TWO strategy conjecture. The condition

$$TWO\uparrow G_c(\mathcal O,\mathcal O)$$

may be added to the list of equivalent conditions in Corollary 4.2. This is presented as a natural conjecture, and its validity is stated to be an open question.