The power-of-two conjecture for odd core partitions into distinct parts

Let $s$ be odd, and consider $(s,s+2)$-core partitions into distinct parts. Power-of-two conjecture. Their number equals

$$2^{s-1}.$$

The paper presents this as a proposed extension beyond the cases it proves, seeking a general enumeration of $(s,t)$-core partitions into distinct parts for coprime $s$ and $t$.