The improved uniform bound for families with bounded symmetric-difference VC-dimension

Let $d<n$ be positive integers with $d ot\bequiv r \pmod 2$ for some $r\in\{0,1\}$. Let $\mathcal F$ be a uniform family of subsets of $[n]$ such that the VC-dimension of its symmetric-difference family satisfies

$$\operatorname{Vdim}(\mathcal F\mathbin{\Delta}\mathcal F)\leq d.$$

Uniform Dvir–Moran conjecture. One should have

$$|\mathcal F|\leq 2^r\binom{n-r}{\lfloor d/2\rfloor}.$$

The paper presents this as the best form of its preceding theorem, refining the general bound for uniform families with bounded symmetric-difference VC-dimension. The statement is given as a conjecture in the concluding remarks, and no resolution is supplied in the source.