Let $Gr(k,V)$ be the Grassmann bundle of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ equipped with a full flag $0=E_0\subset E_1 \subset \ldots \subset E_{n-1}\subset E_n=V$. Consider the Schubert variety $$X_w=\{V_k \in Gr(k,V) \ \vert \ \dim(V_k\cap E_{n-k})\geq q\}$$ for some $0 \leq q \leq \min(k,n-k)$. It is known that $X_w$ is Gorenstein (e.g. Nicolas Perrin, "Small resolutions of minuscule Schubert varieties").
Now we have a standard resolution $$\pi: \tilde{X}_w=\{V_q \subset V_k \ \vert \ V_q \subset E_{n-k}\} \longrightarrow X_w, \qquad (V_q, V_k) \mapsto V_k)$$ of the Schubert variety. This $\tilde{X}_w$ can be viewed as a tower of two Grassmannians, namely a Grassmannian $Gr(k-q, V/V_q)$ over the Grassmannian $Gr(q,E_{n-k})$. It is thus possible to compute the canonical bundle of $\tilde{X}_w$ by well-known means (since $\omega_{Gr(k,n)}= \mathcal{O}(-n)= \det(S_k)^{\otimes n}$).
My questions are the following:
Can one identify $\pi$ with the blowup of the Schubert variety $X_w$ along the subvariety $$Z=\{V_k \in X_w \ \vert \dim(V_k \cap E_{n-k}) \geq q+1\}\subset X_w$$ which is of codimension $2q+1$ (inside $X_q$) and also Gorenstein?
At the end of the day I am interested in the canonical bundle $\omega_{X_w}$ which lives in $\mathrm{Pic}(X_w) \cong \mathbb{Z} \det(S_k)$ (since $X_w$ is Gorenstein) where $S_k$ denotes the rank-$k$-tautological bundle over $Gr(k,n)$. I was hoping to get there by using the resolution $\tilde{X}_w$ and the identity $\omega_{\tilde{X}_w}=\det(S_q)^{\otimes q-k} \otimes \det(S_k)^{\otimes n-q} \in \mathrm{Pic}(\tilde{X}_w) = \mathbb{Z}\det(S_q) \oplus \mathbb{Z} \det(S_k)$ which immediately follows from the description of $\tilde{X}_w$ above and the formula for $\omega_{Gr}$. Is there any way to compute this canonical bundle?