The extremal complement conjecture for independent and connected sets

Let $d$ be a positive integer and let $n$ satisfy

$$d+1\le n\le 2d.$$

For a graph $G$, write $i(G)$ for its number of independent sets, $N(G)$ for its number of connected sets, and $N_{dom}(G)$ for its number of dominating connected sets. Let $K_{n-d}$ denote the complete graph on $n-d$ vertices. Extremal complement conjecture. Among all $d$-regular graphs of order $n$, for each of $i(G)$, $N(G)$, and $N_{dom}(G)$, a maximizing graph is the complement of one or more copies of $K_{n-d}$ together with a graph $H$ satisfying

$$\lvert H\rvert<2(n-d).$$

The graph $H$ may depend on which quantity is being maximized. This conjecture is presented as an analogue of an earlier conjecture concerning the number of cliques in complements. It predicts a common extremal structure for three graph-counting problems in the range $d+1\le n\le 2d$; the source supplies no resolution, so its status remains open.