On intersection density of transitive groups of degree a product of two odd primes

Two elements $g$ and $h$ of a permutation group $G$ acting on set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, subset ${\cal F}$ is an every pair intersecting. The intersection density $\rho(G)$ transitive the maximum value quotient $|{\cal F}|/|G_v|$ where $G_v$ stabilizer $v\in V$ runs over all sets in $G$. Intersection densities groups degree $pq$, $p>q$ odd primes, considered. In particular, conjecture that such equal $1$ (posed [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) disproved by constructing family imprimitive $pq$ (with blocks size $q$), $p=(q^k-1)/(q-1)$, whose $q$. construction depends heavily certain equidistant cyclic codes $[p,k]_q$ field $\mathbb{F}_q$ codewords have Hamming weight strictly smaller than $p$.