The extremal susceptibility conjecture for regular graphs

Let $d\geq2$ and $\lambda>0$. Let $\mathfrak{G}(n,d)$ be the collection of all connected $n$-vertex $d$-regular graphs, and let $H_{d,n}$ be the graph defined in the source by arranging copies of the graph obtained from the complete graph on $d$ vertices by deleting one edge into a cycle and joining consecutive copies by one edge. The extremal susceptibility conjecture. For fixed $d$ and $\lambda$,

$$\max_{G\in\mathfrak{G}(d\lceil n/d\rceil,d)}\mathbb{E}_{\lambda}[\mathcal{S}(G)]=(1+o(1))\mathbb{E}_{\lambda}[\mathcal{S}(H_{d,n})].$$

Moreover, if $(d_n)_{n\in\mathbb{N}}$ diverges with $d_n\leq n$ for every $n$, then for every sequence $(\lambda_n)_{n\in\mathbb{N}}$,

$$\max_{G\in\mathfrak{G}(d_n\lceil n/d_n\rceil,d_n)}\mathbb{E}_{\lambda_n}[\mathcal{S}(G)]=(1+o(1))\mathbb{E}_{\lambda_n}[\mathcal{S}(H_{d_n,n})].$$

This conjecture identifies the explicitly constructed graphs $H_{d,n}$ as asymptotically extremal for expected susceptibility among regular connected graphs with the corresponding number of vertices and degree. The source motivates it with examples showing large susceptibility, but provides no resolution.