Shapiro–Shapiro conjecture for totally positive flags

Let $m,p>1$, let $\operatorname{Grass}(p,m+p)$ be the Grassmannian, and let $\alpha^1,\ldots,\alpha^n$ be Schubert data. A real upper-triangular matrix with diagonal entries $1$ is totally positive when every minor not forced to vanish by triangularity is positive; this induces an order $F^1<\cdots<F^n$ on real complete flags. Shapiro–Shapiro conjecture. If $F^1<\cdots<F^n$ are real flags, then the Schubert varieties $\Omega_{\alpha^1}F^1,\ldots,\Omega_{\alpha^n}F^n$ intersect transversally, and all points of the intersection are real. The conjecture generalizes the osculating-flag construction to totally positive flags; the paper proves the first nontrivial instance and gives computational evidence, but does not establish the general claim.