The $(3,n)$ Schur-coefficient symmetry conjecture

For nonnegative integers $a,b,n$, let $[s_{2^a1^b}]_{3,n}$ denote the coefficient of the Schur function $s_{2^a1^b}$ in the relevant Schur expansion for the $(3,n)$ case. The $(3,n)$ symmetry conjecture. For all $a,b,n\geq 0$,

$$[s_{2^a1^b}]_{3,n}=[s_{2^b1^a}]_{3,3(a+b)-n}.$$

This symmetry is presented as following from the conjectured switch bijection in the coprime case; its status beyond that setting is not established in the supplied text.