Completeness conjecture for triangles satisfying Pick's formula almost correctly

Let $p,q \in \mathbb{Z}$ be coprime integers with $1<q<p^2$, with $p$ odd and $q$ even. Let $\Delta(p,q)$ denote the triangle defined in the paper, and suppose that Pick's formula is almost correct for every dilate $r\Delta(p,q)$ with $r\in\mathbb{N}$. Completeness conjecture. The pair $(p,q)$ belongs to one of the four Casson–Gordon families stated above. This conjecture asserts that the known Casson–Gordon families exhaust all pairs for which Pick's formula is almost correct for every dilate; the paper presents empirical evidence but does not establish completeness.