First sheaf cohomology of closed 1-forms on a homogeneous space

Question

Given a smooth quasi-projective variety $X$ over $\mathbb C$, is there a good way to compute the first sheaf cohomologiy $H^1(X,\Omega_{X,cl}^1)$ of closed 1-forms $\Omega_{X,cl}^1$?

What I'm mostly interested in is the case where $X = G/P$ is a homogeneous space of an algebraic group $G$, where $P$ is a closed subgroup. The case where $G$ is reductive and $P$ is a parabolic subgroup (so $X$ is projective) is my primary concern, but it'll be nice if a description for all algebraic groups can be found.

Motivation: $H^1(X,\Omega_{X,cl}^1)$ parametrizes all twisted sheaves of differential operators on $X$.

Some thoughts in the case where $G$ is reductive and $P = B$ is a Borel subgroup that may or may not help:

• We know $H^i(X,\Omega_X^1) = 0$ if $i \neq 1$ and is $\mathbb{C}^{\mathrm{rank} G}$ if $i =1$. So the long exact sequence induced by $0 \to \Omega_{X,cl}^1 \to \Omega_X^1 \to \Omega_X^1/\Omega_{X,cl}^1 \to 0$ reduces to $$0 \to H^0(X,\Omega_X^1/\Omega_{X,cl}^1) \to H^1(X,\Omega_{X,cl}^1) \to H^1(X,\Omega_X^1) \to H^1(X,\Omega_X^1/\Omega_{X,cl}^1) \to H^2(X,\Omega_{X,cl}^1) \to 0,$$ but we need more knowledge of cohomologies of $\Omega_X^1/\Omega_{X,cl}^1$ to proceed.
• Homogeneous twisted sheaves of differential operators (htdo) are parametrized by $\mathfrak{b}/[\mathfrak{b},\mathfrak{b}]$ which has dimension $\mathrm{rank}G$. Therefore $H^1(X,\Omega_{X,cl}^1) \to H^1(X,\Omega_X^1)$ cannot be a proper injection.
• If we view $X$ as the variety of all Borel subalgebras of $\mathfrak g$, and let $\mathcal{N} \subset \mathcal{B} \subset X \times \mathfrak g$ to be subbundles of $X \times \mathfrak g$ whose fibers at a point $x$ (which corresponds to a Borel subalgebra $\mathfrak b_x$) are $[\mathfrak{b}_x,\mathfrak{b}_x]$ and $\mathfrak b_x$, respectively, then we have an exact sequence of vector bundles $0 \to \mathcal B \to X \times \mathfrak g \to TX \to 0$ and the corresponding sequence $0 \to \mathfrak b^\circ \to \mathfrak g^\circ \to \mathcal T_X \to 0$ of sheaves; similarly if $\mathfrak n^\circ$ is the sheaf of $\mathcal N$ then $$\Omega_X^1 \cong \mathfrak n^\circ = \big\{\sum_i h_i \otimes \xi_i \in \mathcal O_X \otimes_{\mathbb C} \mathfrak g = \mathfrak g^\circ \mid \sum_i h_i(x) \otimes \xi_i \in [\mathfrak{b}_x,\mathfrak{b}_x] \big\}.$$ Translating the pairing $\mathcal T_X \otimes_{\mathcal O_X} \Omega_X^1 \to \mathcal O_X$ and the definition of exterior derivative on $\Omega_X^1$, one can write down a condition for a section $\sum_i h_i \otimes \xi_i \in \mathfrak n^\circ$ to be closed, namely that for any $\eta_1,\eta_2 \in \mathfrak g$ (viewed also as global vector fields on $X$), the function $$\sum_i \Big( B(\xi_i,\eta_2) \eta_1(h_i) - B(\xi_i,\eta_1) \eta_2(h_i) - B(\xi_i,[\eta_1,\eta_2]) h_i \Big) \in \mathcal O_X$$ vanish, where $B(-,-)$ is the Killing form. But I don't know how to proceed.

Thanks in advance for any help.