Edel–Landjev–Xiang conjecture on the highest intersections with Hermitian surfaces

Let $V_2\subset\mathbb{P}^3(\mathbb{F}_{q^2})$ be a non-degenerate Hermitian surface, and let $F\in\mathbb{F}_{q^2}[x_0,x_1,x_2,x_3]$ be homogeneous of degree $d$. Let $w_i$, for $1\leq i\leq 2d+1$, denote the $2d+1$ highest possible values of

$$|V(F)\cap V_2|.$$

Edel–Landjev–Xiang conjecture. For each $i$, there exist linear homogeneous polynomials $H_1,\ldots,H_d$ such that the hyperplanes $V(H_1),\ldots,V(H_d)$ contain a common line and

$$|V(H_1\cdots H_d)\cap V_2|=w_i.$$

The conjecture describes the extremal intersection values by unions of $d$ hyperplanes through a common line. It is stated here as a specialization of Conjecture 2(i) from the cited work, and no resolution is supplied in the text.