The completeness conjecture for Constructions A, B and C

Let $PG(2,q)$ be the projective plane of order $q$. Constructions A, B, and C produce families of complete $k$-arcs in the size regions stated in the paper's theorem: Construction A for prime $q$, Construction B for prime powers $q\not\equiv3\pmod 4$, and Construction C for prime powers $q\equiv3\pmod4$, with the corresponding bounds on $k$ and the listed verified exceptional ranges. Completeness conjecture for Constructions A, B and C. The assertions of that theorem hold also for every $q>1367$. This would extend the experimentally verified completeness of the three construction families beyond the ranges proved in the paper.