Breaker’s conjecture for doubly-biased component games

Let $G$ be a $d$-regular graph on $n$ vertices, where $d\ge3$, and let $m$ be a positive integer. In the $(m:(d-2)m)$ game on $G$, Breaker’s conjecture. Breaker can force Maker to build only connected components of size $o(n)$, perhaps polylogarithmic or even logarithmic in $n$. This asks whether Breaker can still prevent large Maker components at the critical bias, where the strategy discussed in the paper is inadequate; the conjectured bound is left open.