Before revision:  It surprisingly turns out that one of such equivalent conditions applies the semidefinite programming (SDP) , which leads to a practical usefulness. As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 (  2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
After revision:  It surprisingly turns out that one of such equivalent conditions applies the semidefinite programming (SDP) J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 ( 2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.