The asymptotic one-point conjecture for affine-flat intersection statistics

Let $d\geqslant 1$, and let $\lambda^*(d,1)$ denote the optimal proportion of affine $d$-flats with intersection size $1$ in the paper's extremal problem. The one-point asymptotic conjecture. We have

$$\lambda^*(d,1)=(1+o_d(1))\cdot\frac{1}{e}.$$

This gives the conjectured asymptotic value in the one-point case and is consistent with the proposed Poisson optimal construction. Existing bounds leave a gap for odd intersection sizes, so the conjecture remains open.