Sørensen's conjecture on Hermitian surface intersections

Let $q$ be a prime power, let $F\in\mathbb{F}_{q^2}[x_0,x_1,x_2,x_3]$ be homogeneous of degree $d$, and let $V_2\subset\mathbb{P}^3$ be a non-degenerate Hermitian surface defined over $\mathbb{F}_{q^2}$. For $d\leq q$, Sørensen's conjecture.

$$\left|(V(F)\cap V_2)(\mathbb{F}_{q^2})\right|\leq d(q^3+q^2-q)+q+1.$$

Moreover, equality holds precisely for surfaces defined by a homogeneous polynomial whose zero locus is a union of $d$ planes in $\mathbb{P}^3(\mathbb{F}_{q^2})$, each tangent to $V_2$, all containing a common line $\ell$ that intersects $V_2$ in $q+1$ points. The conjecture gives the maximal number of rational points in such intersections and determines the minimum distance of the associated codes; the paper's abstract states that it proves Sørensen's conjecture.