Has anyone had experience with mortgage brokers negotiating a better rate after finding a different offer?

Hi all,

We just purchased our first home over the weekend! Our mortgage broker has been incredibly helpful throughout the process. They presented us with a few options for lenders, ultimately recommending ING with a 6.14% interest rate.

A day later, a lender from ANZ bank contacted me directly and offered a slightly better deal: 6.17% interest with $3,000 cashback for first-time homebuyers.

I informed my mortgage broker about the ANZ offer, and they're checking to see if they can match it. The ANZ lender did mention to me the above rate of 6.17% is only valid if i went with the him.

What are the chances my mortgage broker can get me the ANZ deal?