The combinatorially symmetric P-matrix completion problem

An n-by-n real matrix is called a P-matrix if all its principal minors are positive. The P-matrix completion problem asks which partial P-matrices have a completion to a P-matrix. Here, we prove that every partial P-matrix with combinatorially symmetric specified entries has a P-matrix completion. The general case, in which the combinatorial symmetry assumption is relaxed, is also discussed.