The block-kernel characterization of exponential graph growth

Let $L$ be a transitive permutation group on a set $\Omega$. A proper block $B$ is a block of imprimitivity with $B\neq\Omega$, and let $N$ be the kernel of the action of $L$ on the block system induced by $B$. The notation $N_{(B)}$ denotes the pointwise stabiliser of $B$ in $N$.

Block-kernel characterization. The group $L$ has exponential graph growth if and only if it admits a proper block $B$ such that

$$N_{(B)}\neq 1.$$

This conjecture seeks a characterization of the transitive permutation groups with exponential graph growth. The paper's main theorem proves the forward-type sufficient condition using a nontrivial block and a nontrivial pointwise stabiliser of its complement, but the stated if-and-only-if characterization remains unresolved.