  We present a conjectural formula for the principal minors and the
characteristic polynomial of Gross's regulator matrix associated to a totally
odd character of a totally real field. The formula is given in terms of the
Eisenstein cocycle, which was defined and studied earlier by the authors and
collaborators. For the determinant of the regulator matrix, our conjecture
follows from recent work of Kakde, Ventullo and the first author. For the
diagonal entries, our conjecture overlaps with the conjectural formula
presented in our prior work. The intermediate cases are new and provide a
refinement of the Gross--Stark conjecture.
