The complete-spectrum conjecture for k-avoiding sets

Let $\mathrm{PG}(2,q)$ be the Desarguesian projective plane of order $q$, and let $\mathrm{Sp}(k,q)$ be the set of cardinalities of point sets avoiding $k$-secants. Complete-spectrum conjecture. Suppose that

$$c\sqrt{q}<k<q+1-c\sqrt{q}$$

for some absolute constant $c$. Then

$$\mathrm{Sp}(k,q)=[0,q^2+q+1],$$

i.e., there exists a $k$-avoiding set in $\mathrm{PG}(2,q)$ of every admissible size. Theorem on square orders shows this conclusion when $q$ is a square, and further results give spectra missing at most six admissible values for certain even orders and smaller $k$; the general assertion remains open.