Shifted twisted Yangians and affine Grassmannian islices

In a prequel we introduced the shifted iYangians ${}^\imath Y_\mu$ associated to quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, and constructed the iGKLO representations of ${}^\imath Y_\mu$, which factor through truncated shifted iYangians ${}^\imath Y_\mu^\lambda$. In this paper, we show that ${}^\imath Y_\mu$ quantizes the involutive fixed point locus ${}^\imath W_\mu$ arising from affine Grassmannians of type ADE, and supply strong evidence toward the expectation that ${}^\imath Y_\mu^\lambda$ quantizes a top-dimensional component of the affine Grassmannian islice ${}^\imath\overline{W}_\mu^\lambda$. We identify the islices ${}^\imath\overline{W}_\mu^\lambda$ in type AI with suitable nilpotent Slodowy slices of type BCD, building on the work of Lusztig and Mirkovi\'c-Vybornov in type A. We propose a framework for producing ortho-symplectic (and hybrid) Coulomb branches from split (and nonsplit) Satake framed double quivers, which are conjectured to relate closely to the islices ${}^\imath\overline{W}_\mu^\lambda$ and the algebras ${}^\imath Y_\mu^\lambda$.