filtfilt initial condition calculation ... can someone explain it?

Can someone explain how to get from the article "Determining the initial states in forward-backward filtering, IEEE Transactions on Signal Processing" to the initial condition calculation used in the filtfilt function?

As far as I understand the article, the calculation works by finding inital values that minimize the difference between forward-backward and backward-forward application of the filter.

But how does this translate into the formula used in filtfilt and the extrapolation by reflection?

Could the extrapolation by reflection be replaced by chosing different initial conditions? The initial conditions of the filter reflect all the memory it has of previous samples.