The rise of capitalism, scientific inquiry, technology and nasty germs made western Eurasian societies so powerful and innovative that people of other continents
could not even hope to rival them for the scale of conquest.
Option Elimination -
A. could not even hope to rival them for - ok
B. were not even their rivals for - rules out the rivalry part.
C. were not even their rivals as to - "as to" is used to clarify a topic or subject in a sentence.
"as to" for introducing a topic.
1. As to the issue of budget cuts, we need to consider alternative solutions.
2. As to the question of whether to expand the company, the board is divided.
"as to" for expressing an opinion or providing information.
1. I don't know as to her whereabouts.
2. The expert provided insights as to how to improve efficiency.
D. could even hope to be its rival in - "its" referring to plural societies - Wrong.
E. were not even hoping to be its rival for - the same "its" issue. Usage of past progressive is unnecessary.