I know she can see 6 or 8. She knows I can see 10 or 12. She knows that I know she can see 6, 8 or 10.

Given this information, we both know that the minimum number she can see that the other knows is 6. This allows us to both shortcut the process. We can both discard the first rounds that eliminate that she can see 2 or 4. The first round we need to play is the one where she can see 6.

So, in my first round, I'd give an answer if I could see 14. I don't give an answer, so she knows I don't see 14. We both already knew that, but it was the first round we could agree on, so that's where we synchronize.

After that, you just play out the rounds as in the proposed solution.

Edit: doesn't work: if she had actually seen 6 rather than 8, she'd have known I'd see 12 or 14, so she'd know I knew she would see 4 , 6 or 8. So you can't synchronize on 6 after all.