Moore graph extremality conjecture for the connected-set growth constant

Let $c_d(n)$ denote the exponential growth constant for the maximum number of connected vertex subsets among $d$-regular graphs of order $n$. A Moore graph is a $d$-regular graph of a given girth that attains the theoretical lower bound, the Moore bound, on its order. If $G$ is a $d$-regular Moore graph of order $n$, then Moore graph extremality conjecture. For every $n'>n$,

$$c_d(n')<c_d(n)=c_d(G).$$

The conjecture asserts that Moore graphs are extremal and that increasing the order beyond a Moore graph strictly lowers the corresponding growth constant. The paper notes that exact extremal graphs are difficult to describe in general and that $c_d(n)$ is not monotone decreasing, so the claim is a specific strict comparison after an order attained by a Moore graph; no resolution is supplied.