Andrews' unboundedness conjecture for the coefficients of the mock theta function sigma

Let

$$\sigma(q)=\sum_{n=0}^\infty \frac{q^{n(n+1)/2}}{(-q;q)_n}=:\sum_{n=0}^\infty S(n)q^n,$$

where $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$. Andrews' unboundedness conjecture.

$$\limsup |S(n)|=+\infty.$$

The coefficients $S(n)$ arise from the mock theta function studied by Andrews, Dyson, and Hickerson and have a partition-theoretic interpretation. The source does not provide evidence that this conjecture has been resolved.