The equal-slope conjecture for Cookie Monster losing positions

Let $p^0_i,q^0_i$ and $p^1_i,q^1_i$ denote the coordinates of the losing positions of the forms $\{0,p^0_i,q^0_i\}$ and $\{1,p^1_i,q^1_i\}$, respectively. Here, the pairs are compared in their natural order, and the slope of each graph is represented by the ratio of its second coordinate to its first coordinate. Equal-slope conjecture. On average, the graphs of $p^0_i$ against $q^0_i$ and $p^1_i$ against $q^1_i$ have the same slope, that is

$$\frac{q^0_i}{p^0_i}\approx\frac{q^1_i}{p^1_i}.$$

This is an empirical conjecture about the structure of losing positions in the Cookie Monster Problem; the source provides no proof or resolution.