Conjecture on sharp polynomial bounds for triangles in pseudo-line arrangements

Let $\mathcal{A}$ be an affine arrangement or a projective arrangement of $n$ pseudo-lines. The quantities $a_3(\mathcal{A})$ and $p_3(\mathcal{A})$ denote the numbers of triangular cells in the affine and projective arrangements, respectively, and the bounds stated in the polynomial-bound theorems apply to the corresponding quantity. Sharpness conjecture. The bounds of Theorems~ and~ are reached for any integer $n\geq 21$. This conjecture extends the previously known special case for projective arrangements with $n\equiv 0,4\pmod{6}$; the preceding theorem establishes sharpness for infinitely many values in every residue class modulo $6$, while the claim concerns every $n\geq 21$.