The decision
You've got the argument and the method. What's left is to decide, and I'd like the decision to be one sentence.
I let them finish. When I've jumped in, I stop and hand it back.
Two parts, and both matter.
The first is the finish. The tail as the target, the drop, the half-second, the word on the pad. One conversation a day for a fortnight, then the rest.
The second is the hand-back. Stop mid-sentence. "Sorry, go on." Don't re-take, don't finish it for them, ask for the place if they've lost it, and mean it. This is the part that'll do most of the work, because you'll jump for a long time yet, and the hand-back turns each jump into something the other person can forgive and, in the end, something they stop noticing because it stopped happening.
There's a form of planning that helps with rules like this, and it's been tested many times. Peter Gollwitzer and Paschal Sheeran reviewed ninety-four studies of it in 2006. You decide in advance what you'll do and when, in the form "when this happens, I'll do that", and people who plan that way follow through much more often than people who only hold the intention. So write three.
When I feel the breath in and the lean, I'll write one word on the pad and look at their mouth until it stops.
When I hear my voice while theirs is still going, I'll stop mid-word and say "sorry, go on".
When they've finished, I'll count one before I speak.
What to expect. This is my expectation and not a finding. The first few days, you'll jump as often as ever and catch about half of them, and the hand-back will feel like stumbling. By the end of the first week the sign will start arriving before the voice, and you'll have a few clean finishes, and you'll notice the tails. Some of them will surprise you. Around the second week, someone will react. Your partner will tell a story to the end, and it'll be a longer story than you've heard from them in a while. A colleague will say something in a meeting they'd been sitting on. By the end of a month the tally will have dropped to a fraction of what it was and you'll have stopped keeping it. You'll still jump. Everyone does. But the jumps will be occasional and handed back, and the people around you will have started bringing the tail again.