Ferris Wheel score conjecture for Dots and Boxes

Let $W_n$ be a wheel graph and let $C(n,n)$ be the corresponding Ferris Wheel graph, for $n\geq 3$. Suppose Player X wins on $W_n$ with score $(a,b)$. Ferris Wheel score conjecture. If Player X is Player 1, then the score on $C(n,n)$ is $(a-1,b)$; if Player X is Player 2, then the score is $(a,b-1)$. The source reports this pattern from computational results and does not give a proof or resolution.