The bounded-difference conjecture for Cookie Monster losing positions

Let $p^0_i,q^0_i$ and $p^1_i,q^1_i$ enumerate, respectively, all losing positions of the forms $\{0,p^0_i,q^0_i\}$ and $\{1,p^1_i,q^1_i\}$, with each family arranged in increasing order of its first variable, $p^0_i$ or $p^1_i$. Bounded-difference conjecture. There exist constants bounding both

$$p^1_i-p^0_i$$

and

$$(q^1_i-p^1_i)-(q^0_i-p^0_i).$$

This conjecture predicts a uniform correspondence between the two ordered families of losing positions, but the source gives no evidence of a proof or resolution.