For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{ D}_k$ such that the category of finite dimensional $\mathbb{ D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${ \rm D}_k$ or ${ \rm B}_{ k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
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