Interlacing of zeros between parameters φ and 1+φ for hypergeometric B_φ

Prove that, for φ ∈ (0, ∞)^N and for A in the normalized Laguerre–Pólya class, the zeros of the n-th Brenke polynomial pn(x) generated by A and associated to B_φ(z) = 0F_N(-; φ1,...,φN; z) interlace in the following precise way between parameter sets φ and 1+φ (the vector obtained by adding 1 to each component): for negative zeros, ζ_{n−}(1+φ) < ζ_{n−}(φ) < ζ_{n−−1}(1+φ) < ... < ζ_{1−}(1+φ) < ζ_{1−}(φ); and for positive zeros, ζ_1(φ) < ζ_1(1+φ) < ζ_2(φ) < ζ_2(1+φ) < ... < ζ_n(φ) < ζ_n(1+φ).

Background

Theorem 6.3 proves strict interlacing of zeros across consecutive degrees for fixed φ. The authors further posit interlacing across parameter shifts, comparing the zeros for φ and 1+φ, which, if true, would give a strong structural relation of zero locations under unit parameter increments.

This interlacing would complement the conjectured monotonicity and invariance properties for the same family B_φ.