Kovács–Nagy density conjecture for [k,t]-flats in binary affine spaces

Let $\mathrm{AG}(n,2)=\mathbb{F}_2^n$, and let $S\subseteq\mathbb{F}_2^n$ be a point set of size $m$. A $[k,t]$-flat is a $k$-dimensional affine subspace containing exactly $t$ points of $S$. Write $[n,m]\to[k,t]$ when every $m$-element subset of $\mathbb{F}_2^n$ induces a $[k,t]$-flat, and define

$$\operatorname{Sp}(n,k,t)=\{0\le m\le 2^n:[n,m]\to[k,t]\},\qquad \rho(n;k,t)=\frac{|\operatorname{Sp}(n,k,t)|}{2^n}.$$

Kovács–Nagy density conjecture. For every $k\ge 1$ and $0\le t\le 2^k$, we have

$$\lim_{n\to\infty}\rho(n;k,t)=1.$$

This conjecture asserts that, for each fixed pair $(k,t)$, almost every cardinality $m$ forces a $[k,t]$-flat in sufficiently high-dimensional binary affine space. It was posed as a binary affine-space analogue of an intersection problem and is presented in the source as an open conjecture.