Weak Grünbaum triangle conjecture for proper circle arrangements

Let ${\cal A}$ be a simple digon-free arrangement of $n$ pairwise intersecting circles, and let $p_3({\cal A})$ denote the number of triangular cells. Weak Grünbaum triangle conjecture. Every such arrangement contains at least $2n-4$ triangles:

$$p_3({\cal A})\ge 2n-4.$$

Counterexamples to Grünbaum's conjecture in the broader pseudocircle setting are non-circularizable, so the paper suggests that the bound may remain true for arrangements of proper circles. No resolution is given.