Nonexistence of the 4-vertex condition for odd-order bicirculants

Prove that no n-bicirculant with odd n greater than 5 satisfies the 4-vertex condition.

Background

The paper proves that no locally 3-isoregular n-bicirculant exists for odd n, and 3-isoregular graphs form a subclass of strongly regular graphs satisfying the 4-vertex condition. Establishing the stronger nonexistence statement for all odd n greater than 5 would contribute to a direct proof of the stated consequence concerning simply primitive permutation groups of degree twice a prime.