Closed formula for the resultant of the Padé approximant polynomials

Prove or disprove the conjectured formula for the resultant r_n of the hypergeometric Padé approximant polynomials A_n(U) and B_n(U), namely r_n=(-1)^{n(n+1)/2}(3n)^{-n}\prod_{m=1}^{n-1}((m^2-1/9)/(n^2-m^2))^{n-m}.

Background

In the analysis of small solutions to x3-(t3-1)y3=q, the paper uses only that the resultant of A_n and B_n is nonzero. It then records an explicit product expression as a conjectural formula for that resultant, leaving its validity unresolved.

References

Thus their resultant $r_n \neq 0$ (conjecturally $$r_n=(-1){n(n+1)/2}(3n){-n}\prod_{m=1}{n-1}\left({m2-{1\over 9} \over n2-m2}\right){n-m}$$ by the way).

— Pencils of norm form equations and a conjecture of Thomas, II  (2609.09995 - Amoroso et al., 9 Sep 2026) in Section small solutions, proof of the proposition following equation (p2)