Suppose I have a symmetric matrix $A$ and several diagonal matrices $D_1,D_2,\dots$. Are there any matrix transformations such as $P^\top A P$ so that

$$P^\top A P, P^\top D_1 P, P^\top D_2 P, \dots$$

are either all tridiagonal, or all have minimimal bandwidth in some sense?

If, for example, I only had $D_1$ then solving the generalized eigenvalue problem for the matrix pencil $A, D_1$ would give me a basis that simultaneously diagonalizes both $A$ and $D_1$. Obviously, one basis will not simultaneously diagonalize more matrices in general, but a set of banded matrices would also be pretty nice.

I am aware of Tisseur and Garvey's papers on simultaneous tridiagonalization of two matrices.