On a nonnegative irreducible matrix that is similar to a positive matrix (Q432213)

Let us define a diagonal of an \(n\times n\) matrix \(A\) to be a sequence of \(n\) of its entries such that no two lie in the same row or the same column. \textit{A. Borobia} and \textit{J. Moro} [``On nonnegative matrices similar to positive matrices'', Linear Algebra Appl. 266, 365--379 (1997; Zbl 0887.15019)] raised the following question: Suppose that \(A\) is an \(n\times n\) irreducible (entrywise) nonnegative matrix such that every diagonal of A contains a positive entry. Is \(A\) then similar to a positive matrix? While this question remain open, the case \(n=2\) is easy, and \(n=3\) has been resolved in the affirmative in the above paper. In the article under review the author gives an affirmative answer in the case \(n = 4\). The proof is computationally involved, distinguishes several cases, and uses \textit{T. J. Laffey}'s work [``Extreme nonnegative matrices'', Linear Algebra Appl. 275--276, 349--357 (1998; Zbl 0933.15028)] on extreme nonnegative matrices.