Frankl's chain-free conjecture for families with bounded VC-dimension

Let $n,d,\ell$ be positive integers satisfying

$$n+\ell\ge 2(d+1),$$

and let $\mathcal{F}\subseteq 2^{[n]}$ be a family whose VC-dimension is at most $d$ and which contains no chain of length $\ell+1$.

Frankl's chain-free conjecture.

$$|\mathcal{F}|\le \sum_{i=d-\ell}^{d}{n\choose i}.$$

This extends the Sperner-family problem from antichains to families with bounded chain length while retaining bounded VC-dimension. The source presents it as an open conjecture of Frankl.