Drop out rates and phone numbers

Assume a customer service website of some description.

The company behind this website wants customers to navigate through it and find the answer to their question themselves.

They do not want customers to just ring up customer service and ask a basic question that is already answered. Though of course they will have to provide this option somewhere for those cases that can’t be easily handled with self-service.

Hiding the phone number to encourage customers to use the self-service is very bad UX; though business-demands of reducing call numbers will often push designers towards this dark design of making it very hard to actually get directly in touch.

I wonder; has there ever been any research into just how much having an easily visible phone number will lead to increased calls but improved UX?

This will heavily vary case by case and so many other aspects of the design of the site and service will come into play.

But I’m curious about whether anyone has come across a decent A and B study that attempts to quantify the value of hiding/showing numbers, how much a more visible contact number impacts drop-out rate of self service, etc...

Any pointers towards work remotely in this area would be very welcome.