Exact log-Sobolev constant on odd cycles

Determine the exact value of the log-Sobolev constant β for simple random walk on the n-cycle when n is odd.

Background

For the simple random walk on the n-cycle with even n, the log-Sobolev, modified log-Sobolev, and spectral constants coincide (up to the stated equalities). However, for odd n, despite comparable bounds, the exact value of the log-Sobolev constant β remains unknown, leaving a basic gap even in this classical model.