Counterexamples to Conjecture 25 and Remark 6 of Garrappa–Gerhold–Popolizio–Simon: Global modulus lower bounds in arXiv:2410.11852v4

Conjecture 25 and the question in Remark 6 of Garrappa, Gerhold, Popolizio and Simon, arXiv:2410.11852v4, both have negative answers. For every 10. A Hankel-contour expansion gives this family, while an exact rational certificate supplies the explicit instance (alpha,beta,z)=(3/2,1/2,-115+10i). The Kummer question has the elementary witness M(1/2,-3/2,z)=exp(z)(1-4z/3+4z^2/3).