Diameter extremality conjecture for Bell-type io-decomposable Riordan graphs

Let $G_n$ be an io-decomposable Riordan graph of the Bell type. The Pascal and Catalan graphs are

$$PG_n=G_n(1/(1-z),z/(1-z)),\qquad CG_n=G_n(C(z),zC(z)).$$

Here ${\rm diam}(G)$ denotes the graph diameter. Diameter extremality conjecture. For $n\geq4$,

$$2={\rm diam}(PG_n)\leq {\rm diam}(G_n)\leq {\rm diam}(CG_n).$$

Moreover, $PG_n$ is the only graph in the class of io-decomposable graphs of the Bell type whose diameter is $2$ for all $n\geq4$. This conjecture identifies the Pascal and Catalan graphs as the lower and upper diameter extremizers in this class; the supplied text does not state whether it has been resolved.