Armstrong's average-size conjecture for simultaneous core partitions

Let $s$ and $t$ be coprime positive integers. An $(s,t)$-core is a partition that is simultaneously an $s$-core and a $t$-core; its size is the sum of its parts. Armstrong's conjecture. The average size of an $(s,t)$-core is

$$\frac{(s+t+1)(s-1)(t-1)}{24}.$$

Equivalently, since there are $\frac{1}{s+t}\binom{s+t}{s}$ such cores, the sum of their sizes is

$$\frac{(s+t+1)(s-1)(t-1)}{24(s+t)}\binom{s+t}{s}.$$

The conjecture predicts a remarkably simple statistic for simultaneous core partitions and is the Catalan case of Armstrong's broader conjecture on simultaneous core partitions. The source presents it as a conjecture; no resolution is supplied here.