KLMW conjecture on equations for the affine Grassmannian of $SL_n$

Let $\mathcal{G} r_{SL_n}$ be the affine Grassmannian embedded in the charge-$0$ Sato Grassmannian $\mathcal{S}\mathcal{G} r$, and let $\mathcal{F}$ be the charge-$0$ Fermion Fock space. Let $V(\Lambda_0)\subseteq\mathcal{F}$ be the subspace cut out by the linear shuffle forms. The resulting square of closed embeddings is

\begin{tikzcd} \mathcal{G} r_{SL_n} \arrow[d,\hookrightarrow] \arrow[r,\hookrightarrow] & \mathbb{P}(V(\Lambda_0)) \arrow[d,\hookrightarrow] \\ \mathcal{S}\mathcal{G} r \arrow[r,\hookrightarrow] & \mathbb{P}(\mathcal{F}) \end{tikzcd}

KLMW conjecture. The square is Cartesian. Equivalently, the shuffle equations cut out $\mathcal{G} r_{SL_n}$ inside $\mathcal{S}\mathcal{G} r$. The paper proves this conjecture over an arbitrary base ring, so its status is solved.