Let A and B be two positive definite, real, symmetric matrices. Their eigenvalues are constraint to be $\lambda(A), \lambda(B)$ respectively. I know (from [1]) that the eigenvalues of AB must obey
$$
\lambda^\downarrow(A) \cdot \lambda^\uparrow(B) \prec \lambda(AB) \prec \lambda^\downarrow(A) \cdot \lambda^\downarrow(B)
$$

My question is, given a set of eigenvalues $\lambda(A),\, \lambda(B)$ and $\lambda(AB)$  which satisfies the above, does there necessarily exist an A and B such that they and AB have the desired eigenvalues?

I've found plenty of material on similar inequalities, but nothing that states if every solution of the inequalities can be realised.

  [1]: https://mathoverflow.net/questions/106191/eigenvalues-of-product-of-two-symmetric-matrices/106199#106199?newreg=f1980dddf98c4cc98efa4879e2e7953f