Existence of a subforest preserving game chromatic number

Let $F$ be a forest, and let $\F'\subseteq F$ be a subforest. Write $\Delta(F')$ for its maximum degree and $\chi_{\rm g}(F')$ for its game chromatic number.

Subforest preservation conjecture. There exists a subforest $F'\subseteq F$ such that

$$\Delta(F')\leq \chi_{\rm g}(F)$$

and

$$\chi_{\rm g}(F')=\chi_{\rm g}(F).$$

The conjecture would reduce the study of the game chromatic number of forests to subforests whose maximum degree is bounded by the original game chromatic number. It is stated in the conclusions as an open question; the preceding results provide classification criteria in some degree-restricted cases but do not establish the claim in general.