The asymptotic bound for families forbidding distance one

Let $[n]=\{1,\ldots,n\}$ and let $\mathcal{P}[n]$ denote its power set. A family $\mathcal{A}\subset\mathcal{P}[n]$ forbids distance one if $|A\backslash B|\neq 1$ for all $A,B\in\mathcal{A}$.

Distance-one extremal conjecture. Every family $\mathcal{A}\subset\mathcal{P}[n]$ that forbids distance one satisfies

$$|\mathcal{A}|\leq (1+o(1))\frac{1}{n}\binom{n}{\lfloor n/2\rfloor}.$$

The paper gives a family of at least $\frac{1}{n}\binom{n}{\lfloor n/2\rfloor}$ sets by selecting a residue class of the sum of the elements among the middle-level sets, and suspects that this lower bound is asymptotically best.