Conjectures for maximum intersection density by permutation-group degree

Let $I(n)$ denote the maximum intersection density among transitive permutation groups of degree $n$. For a transitive group $G$ of degree $n$, let $\Gamma_G$ be its derangement graph.

Conjectures for $I(n)$.

1. If $n$ is even but not a power of $2$, then there is a transitive group $G$ of degree $n$ such that $\Gamma_G$ is a complete multipartite graph with $n/2$ parts.
2. If $n$ is a prime power, then $I(n)=1$.
3. If $n=pq$ where $p$ and $q$ are odd primes, then $I(n)=1$.
4. If $n=2q$ where $q$ is prime, then $I(n)=2$.

These conjectures are based on computational evidence for transitive groups of degree at most $48$. The source presents them as conjectural values and constructions for the maximum intersection density.