The asymptotic conjecture for affine-flat intersection statistics with even s

Let $d\geqslant 1$ and let $s=j\cdot 2^k$, where $j$ is odd and $1\leqslant k\leqslant d$. The quantities $\lambda^*(d,s)$ measure the optimal proportion of affine $d$-flats having the prescribed intersection size $s$ in the relevant extremal problem, and $c(d,k)$ is the comparison quantity defined in the paper. The asymptotic conjecture. We have

$$\lambda^*(d,s)=(1+o_d(1))c(d,k).$$

Here the conjecture concerns the regime in which $k$ is fixed and $d\to\infty$. The known bounds determine $\lambda^*(d,s)$ only up to the leading constant in the error term, and proving the conjecture likely requires structural information about extremizers, which is currently unavailable.