Issue on class emcee.moves.GaussianMove(cov, mode='vector', factor=None)

For class emcee.moves.GaussianMove(cov, mode='vector', factor=None), the emcee docs (and the comments in the source code) all say the parameter factor (Optional[float]) is invalid for the "vector" mode. However, I do not find factor option is forbidden in actual calculation of proposal samples when mode="vector" in the following code.
Also, I don't quite understand mathematically why factor (Optional[float])` is invalid for the "vector" mode.

As far as I understand, factor (Optional[float]) makes the Gaussian with flexible scale (i.e., add a factor to the fixed input scale parameter), which somewhat introduce more randomness to benefit the proposal.
Then the proposal distribution turn out to be

Here, x is the proposal sample, G represent Gaussian distribution, cov and factor are the input parameters, f is the proposal factor value in actual calculation, which follows LogUniform(-self.log_factor, self.log_factor) in this line.

From my understanding, the proposal distribution keeps a symmetric distribution, which means we do not need to introduce additional factor to account for difference in proposal distribution q(x|x') and q(x'|x) when calculating the acceptance rate in MCMC, i.e., the same as a regular Gaussian proposal.
And it's nothing related to mode if I understand correctly.

Also, another question comes to me when thinking of the mode. If one provides the full, square covariance matrix and choose "random", or "sequential" mode, that means, sampler selects one dimension in each iteration, proposal distribution becomes marginalized Gaussian in this situation. Only the variance element will affect the proposal but the covariance (non-diagonal elements) won't be considered. That's probably a loss of information. Maybe "random", or "sequential" mode should be disabled in this case?

I may make some mistakes when thinking of all these. Please correct me if there're some problems.
Thanks!

Start with the stable moves documentation and src/emcee/moves/gaussian.py, especially the GaussianMove parameter comments and factor calculation. Reproduce the vector, random, and sequential modes described in the issue, then determine whether the implementation or documentation is inconsistent; done means the behavior and its mathematical explanation are aligned.