Wronski map and positive Grassmannians

The B. and M.Shapiro conjecture states that if the Wronskian of polynomials with complex coefficients have only real roots, then the span of those polynomials has a basis given by polynomials with real coefficients. The conjecture has several important reformulations in real algebraic geometry. The conjecture was proved (also for quasi-exponentials, products of polynomials and exponentials of linear functions) by E.Mukhin, A.Varchenko and myself using the completeness of the Bethe ansatz for the $\mathfrak{gl}_N$ Gaudin model and the symmetry of the Gaudin Hamiltonians with respect to the tensor Shapovalov form.
Recently, S.Karp and K.Purbhoo showed that the span of polynomials in question belongs to the totally positive part of the Grassmannian define by the Taylor expansion at a real point greater than all roots of the Wronskian. In a joint work with S.Karp and E.Mukhin we extended this statement to quasi-exponentials and showed that this positivity statement corresponds to the positivity of higher transfer-matrices of Gaudin model corresponding to polynomial irreducible representations of $\mathfrak{gl}_N$ introduced by A.Alexandrov, S.Leurent, Z.Tsuboi, and A.Zabrodin.