Slavov's algebraic Kakeya lower-bound conjecture

Let $L(t_1,t_2),M(t_1,t_2)\in\mathbb{F}_{q_0}[t_1,t_2]$ be arbitrary polynomials. Consider the map

$$\varphi:\mathbb{A}^3_{\mathbb{F}_{q_0}}\longrightarrow\mathbb{A}^3_{\mathbb{F}_{q_0}},\qquad (s,t_1,t_2)\longmapsto (s,st_1+L(t_1,t_2),st_2+M(t_1,t_2)).$$

For each extension $\mathbb{F}_q/\mathbb{F}_{q_0}$, let $E_{\mathbb{F}_q}$ be the image of the induced map on $\mathbb{F}_q$-points. Algebraic Kakeya lower-bound conjecture. One has

$$|E_{\mathbb{F}_q}|\geq\frac{q^3}{4}-O(q^{5/2}),$$

where the implied constant depends only on the degrees of $L$ and $M$. This framework seeks a uniform algebraic-geometric explanation of Kakeya lower bounds over all finite-field extensions; the conjecture is presented as an open problem, while the paper proves special cases for particular forms of the polynomials.