Resolution conjecture for the singular loci and osculating projections of Grassmannians

Let $R_i$ be the relevant subvariety of $\mathbb G(r,n)$, and for $1\leq i\leq r$ let $\alpha_j$ be the successive blow-ups along the smooth centers introduced in the construction. Let $\tau_i$ be the rational map given by projection from the $i$-th osculating space. Resolution conjecture. For each $1\leq i\leq r$, the singularities of $R_i$ are resolved by the sequence

$$R_i^i\stackrel{\alpha_{i-1}}{\longrightarrow}\cdots\stackrel{\alpha_1}{\longrightarrow}R_i^1\stackrel{\alpha_0}{\longrightarrow}R_i,$$

and the composite sequence of blow-ups

$$\alpha_{i-1}\circ\dots\circ\alpha_1\circ\alpha_0$$

resolves the rational map $\tau_i$. The excerpt introduces these maps and blow-up centers but does not state any resolution result or status beyond this conjectural assertion.