Bondy's reverse degree-sum conjecture for long cycles

Let $G$ be a graph with minimum degree $\Greekmath 010E$ and connectivity $\Greekmath 0114$, where $1\leq\Greekmath 0114 \leq\Greekmath 010E$. Let $C$ be a longest cycle in $G$, let $c=|C|$, and let $\overline{p}$ be the order of a longest path in $G-C$. For a positive integer $r$, let $\Greekmath 011B _r$ denote the minimum degree sum of an independent set of $r$ vertices.

Bondy's reverse degree-sum conjecture. If $\overline{p}\geq\Greekmath 0114 -1$, then

$$c\geq\Greekmath 011B _\Greekmath 0114 -\Greekmath 0114 (\Greekmath 0114 -2).$$

This is the long-cycle analogue of Bondy's path version. The cases $\Greekmath 0114 =1,2,3,4$ have the cited partial results of Dirac, Bondy, Bermond, Linial, Fraisse, Yung, and Chiba, Tsugaki, and Yamashita; the general statement is not reported as resolved here.