Why does the determinant of the Dirac operator on $S^n$ approach $1$ as $n\to\infty$?

Question

Bär and Schopka (reference below) present an interesting conjecture regarding the determinant of the Dirac operator on spheres $S^n$. The conjecture is simple: in the limit $n\to\infty$, the determinant goes to $1$.

The numerical evidence for this conjecture is impressive, but the authors said they "have no explanation for this phenomenon." That was published in 2003.

Has any new insight been gained about why this conjecture might be true (or false)?

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Reference:

Bär and Schopka (2003), "The Dirac Determinant of Spherical Space Forms", in Geometric Analysis and Nonlinear Partial Differential Equations (link to pdf: https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/determinante.pdf). The conjecture is shown on page 16 in the pdf file, after theorem 4.4.

Answer 1

A comment posted by Anthony Carapetis cites

• Møller, "Dimensional asymptotics of determinants on $S^n$, and proof of Bär-Schopka conjecture," Math. Ann. 343 (2009) 35–51, https://arxiv.org/abs/0709.0067

which proves the conjecture.