Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices

Which assignments from 2n−12n−1 arbitrary, distinct real numbers as eigenvalues of designated leading principal submatrices permit a real symmetric tridiagonal matrix? We raise this question, motivated both by known results and recent work on multiplicities and interlacing equalities in symmetric matrices whose graph is a given tree. Known results are reviewed, a general conjecture is given, and several new partial results are proved.