The diamond-free family conjecture for the subspace lattice

Let $\mathcal{L}_{n}(q)$ be the lattice of subspaces of $\mathbb{F}_{q}^{n}$, and let $\mathcal{Q}_{2}$ denote the diamond poset. Write $ex_{q}(n,\mathcal{Q}_{2})$ for the maximum size of a family of subspaces of $\mathbb{F}_{q}^{n}$ containing no copy of $\mathcal{Q}_{2}$. Diamond-free subspace-family conjecture.

$$ex_{q}(n,\mathcal{Q}_{2})=2\genfrac{[}{]}{0pt}{}{n}{\lfloor n/2\rfloor}.$$

The two middle levels provide a lower bound of $(2-o(1))\genfrac{[}{]}{0pt}{}{n}{\lfloor n/2\rfloor}$, whereas the best upper bound stated in the paper has leading constant $(\sqrt{2}+3)/2<2.20711$; the conjectured exact asymptotic remains open.