Prellberg–Stanton overpartition coefficient nonnegativity conjecture

Let $n$ be a positive integer, and write the resulting formal power series in $q$ as a polynomial or power series with coefficients in the relevant coefficient ring. Prellberg–Stanton overpartition conjecture. For all positive integers $n$, the coefficients of

$$(1-q)\frac{(-q^n)_n}{(q^n)_n}+q$$

are non-negative. This conjecture proposes an overpartition analogue of the nonnegativity result of Prellberg and Stanton; the source reports numerical evidence but gives no proof or resolution.