Quantum modularity conjecture for 2-shifted symmetric functions

Let $f$ be a 2-shifted symmetric function of weight $k$, and let $\langle f\rangle_q$ denote its associated $q$-bracket.

Quantum modularity conjecture. The $q$-bracket $\langle f\rangle_q$ is a quantum modular form of weight $k$.

This is proposed as an analogue of the corresponding result for shifted symmetric functions in the complex-origami setting. The supplied text gives numerical evidence for the related series $F_3(q)$ but does not establish the conjecture.