The sharp stability conjecture for the edge-isoperimetric inequality in the discrete cube

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{P}([n])$ denote the discrete cube, and let $\partial\mathcal{F}$ be the edge boundary of a family $\mathcal{F}\subset\mathcal{P}([n])$. Let $\mathcal{L}\subset\mathcal{P}([n])$ be the initial segment of the lexicographic ordering with $|\mathcal{L}|=|\mathcal{F}|$. Two families are weakly isomorphic when one can be obtained from the other by a cube automorphism of the relevant type.

The sharp stability conjecture. If $\mathcal{F}\subset\mathcal{P}([n])$ and $\mathcal{L}\subset\mathcal{P}([n])$ is as above, then there exists a family $\mathcal{G}\subset\mathcal{P}([n])$ weakly isomorphic to $\mathcal{L}$ such that

$$|\mathcal{F}\mathbin{\Delta}\mathcal{G}|\leq 2\bigl(|\partial\mathcal{F}|-|\partial\mathcal{L}|\bigr).$$

The paper proves the same stability statement with an unspecified absolute constant in place of $2$, and explains that the constant $2$ is the conjectured sharp value. The conjecture remains open in the source.