Ramsey implication for biased Maker–Breaker games

Prove that for every positive integer bias b and all graphs G and H, the implication G\rightarrow_{b+1}H\Rightarrow G\,MB_b\,H holds.

Background

The paper proves by strategy stealing that G\rightarrow_2 H implies G MB_1 H. The authors explain that this argument fails directly for biased games, but conjecture that the analogous implication remains valid when the Ramsey coloring uses b+1 colors and Breaker has bias b.