There is amazing experimental evidence for the
Monotone Conjecture.
A recent experimental project involving Ruffo, Sivan, Soprunova, and
Sottile used 15.76 gigaHertz-year of CPU time investigating
Shapiro-type problems on flag manifolds.
In all 525.420.135 instances were computed.
This included the investigation of the
Shapiro Conjecture for Grassmannians
mentioned earlier.
It also included over 2 gigaHertz-years investigation on the
Monotone Conjecture.
The Monotone Conjecture is also true
asymptotically:
If the Grassmannian conditions are that a p-plane meets a
complimentary dimensional osculating subspace, then
every partial flag satisfying such conditions
is real, if the points of osculation are sufficiently close together.
Eremenko, Gabrielov, Shaprio, and Vainshtein's result on
separated secants proves it for flags of type
(n-2,n-1).
