Körner–Malvenuto's maximal colliding-permutations conjecture

Let $[n]=\{1,\dots,n\}$. Two permutations $\pi_1,\pi_2$ of $[n]$ are colliding if there is an index $i\in[n]$ such that $|\pi_1(i)-\pi_2(i)|=1$.

Körner–Malvenuto's conjecture. The maximal number of pairwise colliding permutations of $[n]$ is

$$\binom{n}{\left\lfloor \frac{n}{2} \right\rfloor}.$$

The paper recalls this earlier extremal question as motivation for studying graph-different Hamiltonian paths. Its status is not specified in the source.