In my experience when Carefirst is specifically providing insurance coverage and Not just Administration of claims for a benefit program , it pays more.

Cigna, Aetna have been significantly lower payer amounts.
Personally I've had jobs with various insurers and have found carefirst the most expensive, biggest network, that paid the most for claims. Aetna has been the narrowest network, cheapest premium and decent payment to providers. Cigna was expensive and they didn't pay much out. But alot of this is dependent on what the company benefit plan rules are, so there is too much variation.

My prior career, the doctors I worked for and the billing I saw showed a higher payment consistently from carefirst of Maryland and most of the carefirst bluecross blue shield PPOs out there. The exception being Anthem...which paid very little.