Straub's largest-size conjecture for odd $(s,s+2)$-core partitions

Let $s$ be an odd positive integer. An $(s,s+2)$-core partition is a partition that is simultaneously an $s$-core and an $(s+2)$-core, and its parts are required to be distinct. Straub's conjecture. The largest size of an $(s,s+2)$-core partition into distinct parts is

$$\frac{(s^2-1)(s+3)(5s+17)}{384}.$$

Straub posed this conjecture in the study of simultaneous core partitions; the surrounding paper focuses on the number and largest size of $(2k+1,2k+3)$-core partitions into distinct parts and states that it verifies the two conjectures mentioned. The supplied text does not explicitly give a resolution status for this individual claim, so it is recorded as open.