They will switch providers
Because new providers always underestimate the amount of work involved, and are therefore able to undercut the bid of an established provider who knows exactly how much work it is.
The new provider will believe by bringing in new technology to reduce support headcount they can make up the difference (as if the old provider didn't have access to the same) but when they are still operating at a big loss after the first year or two top management will panic.
With a new mandate to "make this account profitable by any and all means" they'll start getting super picky about charging for all the changes the client wants instead of letting them skate by on smaller changes they could technically charge for but hadn't been, which slowly builds up resentment in the client's management. By the time the contract is up for renewal the incumbent bids the actual cost of delivery and wouldn't be too upset if the client left, and the client - thinking "things will be different this time" - happily switches providers for an undercut bid by a new provider who underestimates the work involved and the cycle begins anew.
Source: too many years of consulting for managed service providers, or for clients of managed service providers, and seeing this same story repeated again and again.