The average-size conjecture for coprime core partitions

Let $a,b\in\mathbb{N}$ be coprime. An $(a,b)$-core is a partition with no hook length divisible by $a$ or $b$; a self-conjugate $(a,b)$-core is one equal to its conjugate. The average size is taken over the finite sets of the relevant cores. Average-size conjecture. The average size of an $(a,b)$-core and the average size of a self-conjugate $(a,b)$-core are both equal to

$$\frac{(a+b+1)(a-1)(b-1)}{24}.$$

Equivalently, since the unique largest $(a,b)$-core is self-conjugate and has size $\frac{(a^2-1)(b^2-1)}{24}$, the average ratio between an $(a,b)$-core and the largest $(a,b)$-core is $\frac{a+b+1}{(a+1)(b+1)}$. Olsson and Stanton proved the asserted uniqueness and maximum-size formula, while the average-size assertion is presented as a conjecture.