Codimension conjecture for irredundant affine hyperplane coverings

Let $q>2$ be a prime power, let $V$ be a vector space over $\operatorname{GF}(q)$, and suppose that $V$ is covered irredundantly by $k$ affine hyperplanes

$$H_1+v_1,H_2+v_2,\dots,H_k+v_k.$$

Here the $H_i$ are the corresponding linear hyperplanes.

Codimension conjecture. The codimension of

$$\bigcap_i H_i$$

is at most $k/(1+\varepsilon_q)$ for some fixed positive constant $\varepsilon_q$ depending only on $q$.

This is the self-contained formulation of the positive-density conjecture for minimal affine hyperplane coverings. The source proves it with $\varepsilon_q\geq\frac12$ for nonprime $q$, while the remaining cases are open.