The Grassmann convexity conjecture by B. Shapiro and M. Shapiro admits
several equivalent formulations.
One of them gives a conjectural formula for the maximal total number
of real zeros of the consecutive Wronskians of an arbitrary
fundamental solution to a disconjugate linear ordinary differential
equation with real time.
Another formulation is in terms of convex curves in the nilpotent
lower triangular group.
There is a very elementary formulation in terms of lists of vectors in $\mathbb{R}^k$.

The conjecture remains open, but several partial results are known.
The formula has already been shown to be a correct lower bound.
It has also been shown to give a correct upper bound in several small
dimensional cases.
More recently, a general explicit upper bound has been obtained.

The aim of this talk is to present the conjecture, state the known
results and present a few proofs.