The left dual exceptional collection conjecture for the $\mathcal X_v$'s

Let $\mathbf G/\mathbf B$ be the generalized flag variety, let $W$ be its Weyl group with longest element $w_0$ and length function $\ell$, and let $\mathcal X_v$ be the objects indexed by $v\in W$ introduced in the paper. Let $\mathcal L(\rho)$ denote the line bundle associated with $\rho$. Left dual exceptional collection conjecture. The left dual exceptional collection to the $\mathcal X_v$'s consists of

$$\mathcal X_{w_0v}\otimes\mathcal L(\rho)[-\ell(w_0v)].$$

This would identify the left dual collection explicitly and provide the expected exceptional-collection structure over the generalized flag variety. The claim is presented as conjectural and is supported by the preceding vanishing and one-dimensional cohomology computations, but no proof or resolution is given here.