Grünbaum's digon conjecture for pairwise crossing pseudocircles

Let a pseudocircle be a simple closed Jordan curve in the plane, and let an arrangement be the cell complex into which the plane is decomposed by a family of pseudocircles. An arrangement is simple if no point lies on three pseudocircles. A digon is a face bounded by exactly two edges.

Grünbaum's digon conjecture. Every simple arrangement of $n>2$ pairwise crossing pseudocircles has at most $2n-2$ digons.

This is the simple-arrangement formulation of Grünbaum's conjecture that the maximum for nontrivial arrangements of pairwise crossing pseudocircles is $2n-2$; the source provides no resolution status.