Characteristic-sequence conjecture for transitive permutations

Let $f$ be a transitive permutation in the symmetric group $S_n$. For each $i=1,\ldots,n-1$, set $A_i=\{i,i+1\}$ and define its characteristic number by

$$m_i=\min\{m\mid (\operatorname{conv}f)^m(A_i)\supseteq A_i\},$$

where, for finite $A\subseteq\mathbb{N}$, $\operatorname{conv}f(A)$ is the convex hull of $f(A)$. Let $m_1'\leq\cdots\leq m_{n-1}'$ be the rearrangement of the numbers $m_1,\ldots,m_{n-1}$.

Characteristic-sequence conjecture. For every transitive $f\in S_n$,

$$m_i'\leq i,\qquad i=1,2,\ldots,n-1.$$

The paper states that this formulation is equivalent to the discrete conjecture above, but the supplied text does not resolve it.