  In this paper, we provide a negative answer to a long-standing open problem
on the compatibility of Spearman's rho matrices. Following an equivalence of
Spearman's rho matrices and linear correlation matrices for dimensions up to 9
in the literature, we show non-equivalence for dimensions 12 or higher. In
particular, we connect this problem with the existence of a random vector under
some linear projection restrictions in two characterization results.
