The optimal-girth conjecture for Wiener-minimizing unicyclic graphs

Let $D=(d_1,d_2,\dots,d_{t-1},d_t,1,\dots,1)$ be a degree sequence with $d_t\geq 2$ and $d_2\neq 2$, and let $G$ be a unicyclic graph with degree sequence $D$. Define

$$D'=(d_1,d_2,\dots,d_{t-1}-1,d_t-1,1,\dots,1)$$

and let $\mathcal{G}(D')$ be the greedy tree with degree sequence $D'$. If $h(\mathcal{G}(D'))$ is achieved by a unique leaf, set $\gamma^*=2h(\mathcal{G}(D'))-1$; otherwise set $\gamma^*=2h(\mathcal{G}(D'))$. Optimal-girth conjecture. The Wiener index of $G$ satisfies

$$W(G)\geq\min\\{\mathcal{G}_u(D,\gamma^*),\mathcal{C}(D,\gamma^*)\\}.$$

This conjecture proposes that, without fixing the girth, a Wiener-minimizing unicyclic graph is captured by one of the two indicated constructions at the girth determined by the height of the associated greedy tree. The supplied text does not state a resolution, so the conjecture is recorded as open.