Tightness of the mindegree-1 Maker-PhantomBreaker lower bound

Let $a,b$ be positive integers such that $\frac{b}{2a}>1$ is an integer. In the $(a:b)$ mindegree-1 Maker-PhantomBreaker game on $E(K_n)$, Maker seeks to claim at least one edge incident with every vertex, while PhantomBreaker claims edges according to the random PhantomBreaker rule of the game.

Mindegree-1 tightness conjecture. Maker has a randomized strategy to win with probability at least

$$(1-o(1))\left(\left(\frac{b}{2a}\right)!\right)^{-1}.$$

The paper states that this bound from Theorem~ is conjectured to be tight. The supplied text does not give evidence resolving the conjecture, so its status remains open.