The order-three conjecture for rational conformal vectors

Let $e$ and $f$ be two distinct rational conformal vectors with central charge $\frac{1}{2}$. The inner product is assumed to satisfy

$$\langle e,f\rangle=\frac{13}{2^8}.$$

Order-three conjecture. Under this assumption, $\tau_e\tau_f$ has order three.

This concerns the relationship between the inner product of two rational conformal vectors and the order of the automorphism generated by their associated $\tau$-involutions. The source presents this as an expectation, and no resolution is given in the supplied text.