Polynomial Sokal conjecture for the roots of P_N

For $N>0$, let

$$P_N(z;q)=\sum_{k=0}^{N}\binom{N}{k}z^kq^{k(k-1)/2}.$$

For $|q|<1$, the polynomial Sokal conjecture. The polynomial $P_N(z;q)$ can have only simple roots, separated in modulus by at least the factor $|q|$. This is the finite-polynomial analogue of the conjecture for the zeros of $F(z;q)$, motivated by the approximation of the rescaled polynomials to the entire function. The supplied text presents it as a conjectural polynomial version and gives no complete resolution.