Weak monotonicity conjecture for the coefficients of $R^{(k)}_1$

Let $R^{(k)}_1$ denote the Bailey-type mock theta family indexed by $k$, and write its Fourier expansion as

$$R^{(k)}_1(q)=\sum_{n\geq 0}a_k(n)q^n.$$

Weak monotonicity conjecture. The coefficients of $R^{(k)}_1$ are weakly increasing for all $k\geq 3$; that is,

$$a_k(n+1)\geq a_k(n)\qquad(n\geq 0).$$

The paper establishes coefficient estimates for the base case of the $R^{(k)}_1$ family and reports numerical evidence for this broader assertion. The conjecture concerns the generalization to all $k\geq 3$, which is left for future work.