Full-spectrum conjecture for affine spaces over $\mathbb{F}_2$

Let $[n,m]\rightarrow[k,t]$ mean that every $m$-element subset of $\mathrm{AG}(n,2)$ has a $k$-dimensional affine subspace containing exactly $t$ points, and let $\rho(n;k,t)$ denote the proportion of integer values $m$ for which $[n,m]\rightarrow[k,t]$ holds. Full-spectrum conjecture. For the relevant fixed integers $k$ and $t$, one has

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

This predicts that almost all set sizes force a $[k,t]$-flat as the dimension tends to infinity; the surrounding results establish this in several cases, while the general assertion is left as a conjecture.