The SL(2) infinite-series conjecture for unitals

Let $k$ be the cardinality of a finite field, let $G=SL(2,\mathbb F_k)$, and let a $G$-space split into one orbit of size $|G|$ and $k+1$ orbits of size $1$. The SL(2) infinite-series conjecture. For every finite field $\mathbb F_k$, there exists a Steiner system (and unital)

$$S(2,k+1,k^3+1)$$

based on this $G$-space. The conjecture has been confirmed for $k\in\{2,3,4,5,8\}$. These designs can correspond to the unitals in Hall planes; for $k\in\{3,4\}$, the corresponding unitals are isomorphic to Hall plane unitals. The assertion remains open for general finite fields.