The Schubert calculus conjecture on transversality of osculating Schubert intersections

Let $Gr_{N+1}({\rm Poly}_d)$ be the Grassmannian of $(N+1)$-dimensional subspaces of the space of complex polynomials of degree at most $d$. Let $\mathcal{F}_\bullet(\xi)$ be the osculating flag at $\xi\in\mathbb{C}\cup\infty$, and fix partitions ${\bf w}(1),\ldots,{\bf w}(n),{\bf w}(n+1)$ satisfying

$$|{\bf w}(1)|+\cdots+|{\bf w}(n+1)|=(N+1)(N-d).$$

For distinct complex numbers $z_1,\ldots,z_n$, write $z=(z_1,\ldots,z_n)$ and

$$\mathcal{I}_{\{{\bf w}\}}(z):=\Omega_{{\bf w}(1)}\left(\mathcal{F}_\bullet(z_1)\right)\cap\cdots\cap\Omega_{{\bf w}(n)}\left(\mathcal{F}_\bullet(z_n)\right)\cap\Omega_{{\bf w}(n+1)}\left(\mathcal{F}_\bullet(\infty)\right).$$

The Schubert calculus conjecture. For generic $z=(z_1,\ldots,z_n)$, the intersection of Schubert varieties $\mathcal{I}_{\{{\bf w}\}}(z)$ is transversal. Here generic means that $z$ does not belong to a suitable proper algebraic subset of $\mathbb{C}^n$. Transversality is known when at least $n-1$ of the partitions are special, but the general case remains open.