The recursive formula for $(3,n)$ Schur coefficients

Let $a,b$ be nonnegative integers with $a<b$, and let $[s_{2^a1^b}]_{3,n}$ denote the coefficient of $s_{2^a1^b}$ in the Schur expansion for the $(3,n)$ case. Write $[r]_{q,t}$ for the corresponding $q,t$-analogue used in the paper. The recursive coefficient conjecture.

$$[s_{2^a1^b}]_{3,n}=(qt)[s_{2^a1^{b-3}}]_{3,n-3}+\sum_{i=0}^{a}[b+i]_{q,t}.$$

The formula is proposed for the coefficients with $a<b$ as a way to compute the remaining $(3,n)$ Schur expansions; no proof or resolution is supplied here.