question about "glaubten sie" set off in mid-sentence by two commas: does it refer both backwards and forwards?

[edmont]

Does "glaubten sie", in the sentence below, refer both to what precedes and what follows it? Or only to what follows it? Thank you very much for any assistance.

Demgegenüber, was sich da an Neid udn Eifersucht unter den Kollegen zeigen
würde [if the university were self-managing], glaubten sie, der Verwaltung durch ein übergeordnetes Kultusministerium doch immer noch den Vorzug geben zu müssen.

[tiorthan]

In this case it's only the stuff that follows.

It's "Demgegenüber glaubten sie, ..." with a dependent clause inserted

[edmont]

Thanks very much, tiorthan.