Bar-Lev–Bshouty–Enis conjecture for polynomial variance functions of natural exponential families

Let a one-parameter natural exponential family (NEF) have variance function $v$ on its mean domain. Let $a_0>0$, let $u_1$ be a complex number with positive imaginary part, and let $\bar{u}_1$ denote its complex conjugate. For $n\in\mathbb{N}$, define

$$v(u)=a_0u\left(u-u_1\right)^n\left(u-\bar{u}_1\right)^n.$$

Bar-Lev–Bshouty–Enis conjecture. The pair $\left(v,\left(0,\infty\right)\right)$ is a variance function for all $n\in\mathbb{N}$ if and only if $\operatorname{Re}(u_1)\leq 0$. The paper's title and abstract indicate that this conjecture is resolved, but the supplied status is unknown; the resolution should be checked in the paper.