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In this paper, we build the unfolding approach from acyclicsign-skew-symmetric matrices of finite rank to skew-symmetric matrices ofinfinite rank, which can be regard as an improvement of that in theskew-symmetrizable case. Using this approach, we give a positive answer to theproblem by Berenstein, Fomin and Zelevinsky in \cite{fz3} which asks whether anacyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. Asapplications, the positivity for cluster algebras in acyclicsign-skew-symmetric case is given; further, the $F$-polynomials of clusteralgebras are proved to have constant term 1 in acyclic sign-skew-symmetriccase.

Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials