Circular variance of a mixture of Von Mises distributions

Does anyone know if there is an analytical solution for the (circular) variance of a mixture of Von Mises distributions that all have equal (circular) mean?

Simulation results suggest that the circular variance of such a mixture is equal to the average of the circular variances of the individual components, at least when the mixture weights are the same.

I would appreciate if anyone can point me to a mathematical proof or rejection. Thanks!