The divisible-set union conjecture

Let $r$ be a positive integer. A set of points over $\mathbb{F}_2$ is $2^r$-divisible if every hyperplane meets it in a number of points congruent to its size modulo $2^r$. An $(r+1)$-space is a projective subspace of dimension $r+1$, and $[r+1]_2=(2^{r+1}-1)/(2-1)$ denotes its number of points.

Divisible-set union conjecture. There exists a function $f\colon \mathbb{N}\to\mathbb{N}$ such that every $2^r$-divisible set of $f(r)\cdot [r+1]_2$ points over $\mathbb{F}_2$ is the disjoint union of $f(r)$ $(r+1)$-spaces and

$$\lim_{r\to\infty} f(r)=\infty.$$

This conjecture proposes arbitrarily large values of $r$ for which a sufficiently structured divisible set must decompose into disjoint spaces. The surrounding discussion gives related small-parameter results and computational evidence, but no resolution of this assertion is supplied.