Have these coloring games been solved? In the paper "On the complexity of some coloring games", Bodlaender gives some open questions about the complexity of deciding if player 1 or 2 has a winning strategy in some graph coloring games. Does anyone know if they have been solved?

1) In one game, two players take turns selecting one vertex in a graph and coloring it properly with a color from a fixed finite set. The loser is the first player who is unable to color a vertex. In Schaefer's paper it is shown to be pspace-complete with 1 color and Bodlaender shows it to be pspace-complete with 2 colors but gives no answer with more color. Is it still open?

2) In another variation, the vertices have numbers 1..n. On a player's turn he must properly color the vertex with the lowest number that has not yet been colored. Again, they are using colors from a fixed set and the loser is the first player who is unable to color his vertex. Bodlaender shows it to be pspace-complete for general graphs. He asks who wins on trees, is it known?

Thanks
The answer is yes, for the first game you list! This result was only established in 2019. Here is a link to the paper: Costa et al. 2019
Even more recently, some variants of the first game were proved to be PSPACE-complete. This result can be found here: Marcilon et al. 2019.