OK. I've figured it out.
As it turns out. There is a bug in HoloEverywhere 1.4.0. The issue has already been reported and fixed: https://github.com/ChristopheVersieux/HoloEverywhere/issues/203
Unfortunately the fix has not been merged to the master branch of HE GitHub project. It's sitting calmly on the dev branch most likely waiting for the next release...
So if you have the same problem as me then you should use the dev branch, or fork the repository and cherry-pick THIS commit to your master branch.
The problem with styles not being applied properly to the dialog still persists though.