Fiat-Shamir for $(2n + 1)$
You can find the security proofs for 5-round Fiat-Shamir:


Ming-Shing Chen and Andreas Hülsing and Joost Rijneveld and Simona Samardjiska and Peter Schwabe: From 5-pass MQ-based identification to MQ-based signatures. Asiacrypt 2016. https://eprint.iacr.org/2016/708
Özgür Dagdelen, David Galindo, Pascal Véron, Sidi Mohamed El Yousfi Alaoui, and Pierre-Louis Cayrel. Extended security arguments for signature schemes. Designs, Codes and Cryptography, 78(2):441–461, 2016. https://link.springer.com/article/10.1007/s10623-014-0009-7


Their proof (may) contain (2n+1)-round Fiat-Shamir.