Armstrong's average-size conjecture for $(a,b)$-core partitions

Let $a$ and $b$ be positive integers, and let an $(a,b)$-core partition be a partition with no hook length divisible by either $a$ or $b$. For a partition $\lambda$, write $|\lambda|$ for its size. If $\gcd(a,b)=1$, then Armstrong's conjecture.

$$\sum_{\lambda} |\lambda|=\frac{(a-1)(b-1)(a+b+1)}{24(a+b)}\binom{a+b}{a},$$

where the sum is over all $(a,b)$-core partitions. Equivalently, the average size of an $(a,b)$-core partition is

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

The conjecture predicts the total and average sizes of the finite set of $(a,b)$-core partitions; it was informally proposed by Armstrong in 2011 and later published. The supplied text does not state whether it has been proved or remains open.