Upper-semicontinuity conjecture for orthogonal complex structures

Let $M$ be a smooth manifold equipped with a Riemannian metric $g$. Consider the space of complex structures on $M$ that are orthogonal to $g$, together with its dimension and number of connected components. Upper-semicontinuity conjecture. The dimension and number of complex structures orthogonal to a given metric are upper-semicontinuous functions in terms of the Riemannian metric. This conjecture formalizes the expectation that special Riemannian metrics have the richest moduli of orthogonal complex structures, whereas generic metrics admit none or only isolated structures.