You can read a review of the paper [in Zentralblatt][1], it contains a short description in German.

The review on [MathSciNet][2] is a bit more extensive (but requires a subscription). There is indeed the condition of $n+1$-to-one from a zero-dimensional compact space onto the space itself. There is also a condition on `gratings' (not defined, but presumably special covers of the space).

In [What is a non-metrizable analog of metrizable compacta? (Part I)][3] Pasynkov defines a class of compacta where the dimensions coincide.


  [1]: https://www.zbmath.org/?q=an%3A0108.35605
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=124031
  [3]: https://doi.org/10.1016/j.topol.2011.04.018