Effective-cone conjecture for complete-intersection line varieties

Let $G$ be the Grassmannian and let $\mathcal{E}$ and $\mathcal{K}_d$ be the bundles appearing in the construction of the line varieties, with $H=c_1(\mathcal{O}(1))$ on $\mathbb{P}(\mathcal{K}_d^*)$. Effective-cone conjecture. Assume $n\geq 2d$. Then for some algebraic class

$$\alpha'\in H^{2n-2d-2}(\mathbb{P}(\mathcal{K}_d^*),\mathbb{Q}),$$

and for some small $\epsilon>0$, the class

$$\pi^*(c_{n-d}(S^{n-d-1}\mathcal{E})-\epsilon l^{n-d})+H\alpha'$$

is effective on $\mathbb{P}(\mathcal{K}_d^*)$. Equivalently, the class $c_{n-d}(S^{n-d-1}\mathcal{E})$ belongs to the interior of the cone of degree $2n-2d$ classes $\alpha$ on $G$ such that $\pi^*\alpha+H\alpha'$ is effective on $\mathbb{P}(\mathcal{K}_d^*)$ for some algebraic class $\alpha'\in H^{2n-2d-2}(\mathbb{P}(\mathcal{K}_d^*),\mathbb{Q})$. This is presented as a reformulation of the very-moving/bigness conjecture in the relevant hypersurface case; the supplied source gives no resolution.