Even Number Game for k at least four

Determine whether Player 1 has a winning strategy in the Even Number Game G_k for every k at least four.

Player 1 is known to have a winning strategy for k=2 and k=3. The outcome for all k at least four is explicitly stated to be unknown.

For $k\geq 4$, this is unknown.

\begin{problem} For $k\geq 4$, does Player 1 have a winning strategy in the game $G_k$?

— Open Problems in Mathematical Logic (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 2