The Grassmannian curve conjecture for semiorthogonal decompositions

Let $C$ be a monotone curve in a $k\times l$ rectangle as described in the construction, with intersection points $Q_i$ and associated integers $a_i,b_i,c_i,d_i$. Define blocks $\SB_i$ by

$$\SB_i = \left\{ \begin{array}{l} d_i + i \ge \lambda_1 \ge \dots \ge \lambda_{a_i} \ge i = \lambda_{a_i+1} = \dots = \lambda_k,\\ \lambda_{k+1} = \dots = \lambda_{n-b_i} = 0 \ge \lambda_{n-b_i+1} \ge \dots \ge \lambda_n \ge -c_i \end{array} \right\}.$$

For $X=\operatorname{Gr}(k,n)$, let $\CA_i=\langle\CU^\lambda\rangle_{\lambda\in\SB_i}$. Grassmannian curve conjecture. The collection $\langle\CA_0,\CA_1,\dots,\CA_m\rangle$ is a semiorthogonal decomposition of $\D^b(\operatorname{Gr}(k,n))$, and each component is generated by an exceptional collection. The construction has the expected number of objects, matching the rank of the Grothendieck group of the Grassmannian; the semiorthogonal and exceptional properties are conjectural in the stated generality.