Spectrum conjecture for complete arcs in projective planes

Let $PG(2,q)$ be the projective plane of order $q$, let $\overline{t}_{2}(2,q)$ be the smallest known size of a complete arc, and let $M_q$ be the upper endpoint defined in the source: for even $q$, $M_q=\frac{1}{2}(q+4)$; for odd $q$, $M_q=\frac{1}{2}(q+7)$ in the stated congruence ranges and otherwise $M_q=\frac{1}{2}(q+5)$. A complete $k$-arc is a complete arc with $k$ points. Spectrum conjecture. Let $353\le q\le 2879$ be an odd prime. Then in $PG(2,q)$ there are complete $k$-arcs of all the sizes in the region

$$\overline{t}_{2}(2,q)\le k\le M_q.$$

Moreover, complete $k$-arcs with

$$\overline{t}_{2}(2,q)\le k\le \frac{1}{2}(q+5)$$

can be obtained by the randomized greedy algorithms of the cited works with a new approach to creation of starting data. The claim is presented as a computational prediction for the indicated finite range; the supplied text gives no resolution status beyond the source's assertion.