The q conjecture for second largest complete arcs

Let $q$ be an odd square with $q\neq 9$. A complete arc in ${\mathrm{PG}}(2,q)$ is an arc that is not properly contained in any larger arc. The second largest complete arc is the largest complete arc after the conic-sized examples.

The $\sqrt{q}$ conjecture. The second largest complete arc in ${\mathrm{PG}}(2,q)$ has size

$$q-\sqrt{q}+1.$$

This conjecture concerns the extremal size of complete arcs not attaining the largest possible size. The paper gives examples of arcs of this size that are not contained in a conic and proves upper bounds implying that sufficiently large arcs are contained in a conic in several cases; the asserted exact value remains open here.