Infinitely many cyclic solutions to the Hamilton–Waterloo problem with odd length cycles

It is conjectured that for every pair (ℓ,m)(ℓ,m) of odd integers greater than 2 with m≡1(modℓ), there exists a cyclic two-factorization of KℓmKℓm having exactly (m−1)/2(m−1)/2 factors of type ℓmℓm and all the others of type mℓmℓ. The authors prove the conjecture in the affirmative when ℓ≡1(mod4) and m≥ℓ2−ℓ+1m≥ℓ2−ℓ+1.