The equal-parameter Armstrong conjecture for simultaneous bicores

Let $s$ be a positive integer and let $a,b$ satisfy $0\leq a,b<s$. For parameters $r$ and $c$, write $\mathcal{C}_{(r\,|\,0,c)}$ for the corresponding set of bipartitions, and let the size of a bipartition mean the total number of boxes in its two components. Equal-parameter Armstrong conjecture. If $s$ and $a-b$ are coprime, then the average size of a bipartition in

$$\mathcal{C}_{(s\,|\,0,a)}\cap\mathcal{C}_{(s\,|\,0,b)}$$

is

$$\frac{(s+1)(a(s-a)+b(s-b)+1-s)}{12}.$$

This is proposed as an analogue of Armstrong's conjecture for simultaneous core bipartitions; the case $a=0$ recovers Armstrong's conjecture for ordinary simultaneous cores. The supplied text gives no resolution, so the conjecture is recorded as open.