Restricting their conjecture to Grassmannians shows a relation to part of the static pole placement problem of linear systems theory. This relation is described in more detail in [So98]. While studying the pole placement problem numerically [RS], Rosenthal and Sottile decided to test some instances of the conjecture of Shapiro and Shapiro. Much to their surprise, the computations were all in agreement with the conjecture. (Here is a description of that project.)
In the aftermath of those computations, restricted versions of the conjecture have appeared in print [RS, So97c,HSS]. Also, Sottile distributed two challenges to the systems solving community. One concerned `hypersurface' Schubert conditions (see Section 2), and the second concerned `Pieri-type' Schubert conditions (see Section 3). A spectacular symbolic computation of Faugère, Rouillier, and Zimmermann [FRZ] was in response to these challenges. They verified one instance of the conjecture involving the 462 4-planes meeting 12 3-panes in 7-space. This renewed Sottile's interest in these conjectures and inspired the recent work. To the best of our knowledge, this document and the paper [So98] mark the debut in print of the most general version of the conjecture of Shapiro and Shapiro.