Log Student-t comparing differences in scale

Modeling
1

I am doing a log student t distribution. All nu values are above 2 so both mean and variance exist.
I want to compare the log sigmas between groups, so taking the difference in the posteriors of log sigma_group1 - log sigma_group2.

If this was a gaussian I would take the CV to control for different means or equally for lognormal model I would first convert the sigma to the arithmetic CV sqrt(exp(log_var) - 1), and for gamma use 1/shape .

Although I don’t believe I can apply this to a student T. So would the perferable thing to do is to take the delta of the log sd? But just not interpret it as the arithmetic CV? Cause I want to make sure that the difference I observe in the residuals isnt due to the difference in the means.

brm.formula <- bf(
  log(y) ~ 0 + group +
    (1 | gr(ID, by = group)),

  sigma ~ 0 + group
)

fit <- brm(
  formula = brm.formula,
  data = dat,
  family = student()
)

Thanks,
Hunter