Birationality criterion for residual linear type monomial ideals

Let $k[\mathbf{x}]=k[x_1,\ldots,x_n]$, and let $F$ be a finite set of $q\geq n$ monomials of the same degree $d\geq 2$. Suppose that the ideal $(F)\subset k[\mathbf{x}]$ is of residual linear type, meaning that the defining bihomogeneous ideal of its Rees algebra is generated in bidegrees $( *,1)$ and $(0,*)$. Let the log-matrix of $F$ record the exponents of its monomials, and let its linear syzygy matrix record the linear syzygies among the elements of $F$. Write $k[\mathbf{x}_d]$ for the $k$-subalgebra generated by all monomials of degree $d$. Birationality criterion. The following conditions are equivalent: (i) both the log-matrix and the linear syzygy matrix of $F$ have maximal rank; (ii) the extension $k[F]\subset k[\mathbf{x}_d]$ is birational. Ideals of residual linear type, also called ideals of fiber type, have defining relations generated by those of the symmetric algebra together with the polynomial relations over $k$. The conjecture asks whether the rank conditions on the monomial data and linear syzygies exactly characterize birationality for this class of ideals; its resolution is not supplied in the source.