The general Frobenius decomposition conjecture for flag varieties

Let $n\in\mathbb N$. Define subcategories of $\mathop{\mathrm D\kern0pt}\nolimits^b(X_n)$ by

$$\mathcal A=\langle\mathcal A_k\rangle,\qquad 0<k\leq n-1,$$

with $\mathcal A_0=\langle\mathcal O_{X_n}\rangle$, and for $k<0$ let $\mathcal A_k$ be defined inductively as the left mutation of the subcategory generated by $\mathcal L_{i\omega_1+(k-i)\omega_{n-1}}$ for $0\leq i\leq k$ through the subcategory generated by $\mathcal A_l$ for $0\leq l<k$. Let

$$\mathcal B=\langle\Lambda^i\mathcal E\otimes\mathcal L_{-\omega_1}\rangle,\qquad 0\leq i\leq n-2,$$

and define $\widetilde{\mathcal A}=\langle\widetilde{\mathcal A}_k\rangle$ for $-n+2<k\leq0$, where $\widetilde{\mathcal A}_0=\langle\mathcal L_{-\omega_1-\omega_n}\rangle$, and for $k<0$, $\widetilde{\mathcal A}_k$ is the right mutation of the subcategory generated by $\mathcal L_{(-k+i)\omega_1-i\omega_{n-1}}\otimes\mathcal L_{-\omega_1-\omega_n}$ for $0\leq i\leq-k$ through the subcategory generated by $\widetilde{\mathcal A}_l$ for $k<l\leq0$. The general Frobenius decomposition conjecture. The collection

$$\langle\widetilde{\mathcal A},\mathcal B,\mathcal A\rangle$$

is a semiorthogonal decomposition of $\mathop{\mathrm D\kern0pt}\nolimits^b(X_n)$ satisfying the conditions of Theorem, and the indecomposable summands of ${\sf F}_*\mathcal O_{X_n}$ are the terms of the right dual decomposition to this semiorthogonal decomposition. This conjecture extrapolates the decompositions established in the preceding low-rank cases to general $n$ and predicts the indecomposable summands of the Frobenius pushforward.