Converse of the JPS characterization (Theorem 1.4)

Determine whether the converse of Theorem 1.4 holds: given a normalized formal power series B(z) = sum_{n>=0} b_n z^n with b_0 = 1 such that RB = L-P0 (i.e., the set of all normalized formal power series A whose Brenke polynomials associated to B have only real zeros equals the normalized Laguerre–Pólya class), prove that B must be an entire function of first type in the Laguerre–Pólya class, that B is not a polynomial, and that the log-concavity limit lim_{n→∞} (b_{n-2} b_n) / (b_{n-1})^2 = 1 holds.

Background

The paper studies when Brenke polynomials generated by A and associated to a given B have only real zeros. The JPS class consists of those B for which this real-rootedness holds for exactly the normalized Laguerre–Pólya class L-P0. Theorem 1.4 proves a sufficient direction: if B is in L-PI, is not a polynomial, and its Taylor coefficients satisfy lim_{n→∞} (b_{n-2} b_n)/(b_{n-1})^2 = 1, then RB = L-P0.

The authors conjecture the converse direction, which would fully characterize the JPS class via intrinsic properties of B (membership in L-PI, non-polynomiality, and the asymptotic ratio condition). They also provide a weaker partial result (Theorem 1.6) toward this conjecture.