Linear variance conjecture for zero-free probability generating functions

Let $X\in\{0,\ldots,n\}$ be a random variable with $p_0p_n>0$ and probability generating function $f_X$. Let $\delta>0$ and $R\geqslant1$. If every zero $\zeta$ of $f_X$ satisfies

$$|\operatorname{arg}(\zeta)|\geqslant\delta,$$

and

$$R^{-1}\leqslant|\zeta|\leqslant R,$$

then Linear variance conjecture.

$$\operatorname{Var}(X)=\Omega_{R,\delta}(n).$$

The conjecture asserts that, under these zero-location restrictions, the variance is essentially as large as possible; the source states that it remains of independent interest and motivates the paper's work.