Don't suppose you have the public domain preprint? is it for calls I suppose? I suppose puts are never optimal?Just saw this thread late. It turns out the double exercise boundary happens naturally for American options under negative interest rates. And you may use the integral formulation to solve the exercise boundary, using similar techniques as in the single boundary case (see Andersen an Lake for example). A paper of mine on this has been accepted for publication in JCF (not sure when it will be actually published with the backlog).
Are you using LSM or PDE approach
// 20-30 years ago the interest rate was [12,15] percent.