Existence conjecture for double blocking sets with two long secants

Let $q$ be a prime power satisfying $q\geq 13$ and $q\not\equiv 2\pmod 3$. A double blocking set in $\operatorname{PG}(2,q)$ is a set meeting every line in at least two points; a $(q-1)$-secant is a line meeting it in exactly $q-1$ points. Existence conjecture. There exists a double blocking set in $\operatorname{PG}(2,q)$ of size $3q-1$ admitting two $(q-1)$-secants. The paper proposes this conjecture after refuting Hill's conjecture; the displayed constructions establish examples only for the listed values of $q$, so the general assertion remains open in the source.