Can I Negotiate a Lower Interest Rate by Requesting a Liability Letter?

My current interest rate is 11.24%. My salary has increased and my credit score has improved to excellent.
I’ve actually received an offer from another bank with a lower interest rate. It’s not the best offer, but I’m planning to request a liability letter from my current bank. If they call and ask why I’m leaving, I’ll mention that I have a lower-rate offer and try to negotiate a better rate with my current bank before making a final decision.
Has anyone tried this approach? Did it work?