Baker and Wantz's 2-rank conjecture for Buekenhout-Metz unitals

Let $q$ be an odd prime power, let $\beta$ be a primitive element of $\mathbb F_{q^2}$, and define

$$C_r=\{(1,y,\beta y^2+r)\mid y\in\mathbb F_{q^2}\}\cup\{(0,0,1)\}$$

for $r\in\mathbb F_q$. Set

$$U_{\beta}=\bigcup_{r\in\mathbb F_q}C_r$$

and let $\mathcal U_{\beta}$ be the $2-(q^3+1,q+1,1)$ design whose points are the points of $U_{\beta}$ and whose blocks are the intersections of secant lines with $U_{\beta}$. The 2-rank conjecture. The binary incidence code of $\mathcal U_{\beta}$ has dimension, or equivalently the $2$-rank of $\mathcal U_{\beta}$, equal to

$$\operatorname{dim} C_2(\mathcal U_{\beta})=q^3+1-q.$$

The proposition preceding this conjecture gives the upper bound $\operatorname{dim} C_2(\mathcal U_{\beta})\le q^3+1-q$; the conjecture asserts that this bound is attained for these Buekenhout-Metz unital designs.