  In previous work we introduced a Khovanov multicurve invariant
$\operatorname{\widetilde{Kh}}$ associated with Conway tangles. Applying ideas
from homological mirror symmetry we show that $\operatorname{\widetilde{Kh}}$
is subject to strong geography restrictions: Every component of the invariant
is linear, in the sense that it admits a lift to a curve homotopic to a
straight line in an appropriate planar cover of the tangle boundary.
