When asking for references, it’s useful to keep in mind that some people generally are willing to do a set amount of “work” versus a specific number of references. The easier you make it to recommend your company, the more likely you can keep pulling that lever. In my opinion, a call between the CTO and CIO is likely the least amount of initial work for the CIO, but you might instead approach them with a template and ask that you can keep it “on file” for future references once it is completed. The CIO may ask that you run each reference by them first, or they may not. You can always make the (easier) additional ask of hopping on the phone for particularly large accounts, but I’d treat it the same way as you’d treat a VC asking to talk to your customers: “I appreciate that speaking with other customers is a great way to understand the impact we’re making. However, I’d like to ask that this is the last step in your due diligence, and that if the reference is positive, you’d be ready to pull the trigger.”
Regardless of the instructions you give, I’d imagine that pricing collusion would be possible or probably if they have good rapport. If these are large enterprises, professionalism may win out. Incidentally, with a written recommendation, you don’t really run this risk.