Geometric description of clasps for the affine Grassmannian of $\mathbb{C}^2$

Let $Y({\underline{i}})$ be the affine-Grassmannian variety associated with ${\underline{i}}=({\0},1^k,{\u2})$ for some $k\in\mathbb N$, and let

$$p:Y({\underline{i}})\longrightarrow \overline{Y({\underline{i}})}$$

be the projection forgetting the intermediate lattices. Let ${\sf{P}}^+$ be the clasp categorifying

$$V_{\Lambda_1}^{\otimes k}\xrightarrow{\pi}V_{k\Lambda_1}\xrightarrow{\iota}V_{\Lambda_1}^{\otimes k},$$

and let $C({\sf{P}}^+)\in D^-(Y({\underline{i}})\times Y({\underline{i}}))$ be its convolution kernel. Geometric clasp conjecture. If $m=2$, then the composition

$$p^*p_*:D^-(Y({\underline{i}}))\longrightarrow D^-(Y({\underline{i}}))$$

is induced by the kernel $C({\sf{P}}^+)$. This predicts a geometric realization of all such clasps for $m=2$, contrasting with the failure of the analogous description for $m>2$.