The c3_{k+1} degree-sum conjecture for 2-factors with at most k cycles

Let $G$ be a graph of order $n$, let $k$ be a positive integer, and let $c3_{k+1}(G)$ denote the minimum degree sum over every set of $k+1$ pairwise nonadjacent vertices of $G$. Degree-sum conjecture. If

$$c3_{k+1}(G)\geq n$$

and every independent set $I$ of $G$ satisfies $|I|\leq \u03b4_G(I)-1$, then $G$ has a 2-factor with at most $k$ cycles. This would extend Ore's theorem from a spanning cycle to a 2-factor with a bounded number of cycles under the stated independent-set condition.