Berndt–Kim eventual sign conjecture for partial theta coefficients

Let $b$ be a positive integer, and define

$$f_b(t):=2\sum_{n=0}^\infty (-1)^n q^{n^2+bn},\qquad q=\frac{1-t}{1+t}.$$

Suppose that, as $t\to0^+$,

$$f_b(t)\sim\sum_{n=0}^\infty a_n t^n.$$

Berndt–Kim conjecture. For sufficiently large $n$, the coefficients $a_n$ have the same sign.

This conjecture concerns the eventual sign pattern of the integer coefficients in the asymptotic expansions of generalized partial theta functions. The paper states that it proves this conjecture, resolving the question posed by Berndt and Kim.