The transversality conjecture for monotone Grassmannian Schubert problems

Let $a=(a_1<\dotsb<a_k)$ and $n$ be positive integers with $a_k<n$. Let $W^a$ index Schubert varieties on $\mathbb F\ell(a;n)$, and let $w_1,\dotsc,w_m\in W^a$ be data for a Schubert problem with every $w_i$ Grassmannian. For $t_i\in\mathbb R\mathbb P^1$, let $X_{w_i}(t_i)$ denote the corresponding Schubert variety for the osculating flag at $t_i$. If the points $t_1,\dotsc,t_m$ are monotone with respect to $w_1,\dotsc,w_m$, then the transversality conjecture. The intersection

$$X_{w_1}(t_1)\cap X_{w_2}(t_2)\cap\dotsb\cap X_{w_m}(t_m)$$

is transverse. This is explicitly presented as a stronger conjecture obtained by discarding the reality conclusion from the monotone conjecture; the source gives experimental support but no general proof.