This conecture is false.

Case 2: If all of the edges of the form $ y_ix_{i+1} $ are contained in the same cycle in a 2-factor of $ G_n $, then replacing the edges $ y_ix_{i+1} $ with the edges $ x_iy_i $ converts this 2-factor of $ G_n $ into a 2-factor of $ n $ disjoint copies of the Petersen graph. Hence, when restricted to each $ H_i $, the 2-factor of $ G_n $ consists of a cycle with 5 vertices and a $ x_iy_i $-path containing a total of 5 vertices. These paths must be joined together through the edges of the form $ y_ix_{i+1} $ creating a cycle of length 5n. Hence, in this case the 2-factor of $ G_n $ contains $ n $ cycles of length 5 and one cycle of length 5n (which is odd).

Now, $ G_n $ is a bridgeless cubic graph whose 2-factors contain only odd cycles, but no 2-factor of $ G_n $ contains fewer than $ n+1 $ cycles.


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