Partition-theoretic density formulas: radial Tauberian theory and non-radial instability

Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sumμ_{\mathcal P}(λ)q^{|λ|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. We determine when this happens. R