Bounded-cycle 2-factor conjecture under degree-sum and independent-set conditions

Let $k$ be a positive integer and let $G$ be a graph of order $n$. For an independent set of order $k+1$, define

$$\sigma_{k+1}(G):=\min\left\{\sum_{v\in I}d_G(v)\;\middle|\; I\text{ is an independent set of order }k+1\text{ of }G\right\},$$

when $\alpha(G)\geq k+1$, and set $\sigma_{k+1}(G)=\infty$ otherwise. For an independent set $I$ of $G$, let $\delta_G(I)$ be the minimum degree of a vertex in $I$. A 2-factor is a $2$-regular spanning subgraph. Bounded-cycle 2-factor conjecture. If

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

and every independent set $I$ of $G$ satisfies

$$|I|\leq \delta_G(I)-1,$$

then $G$ has a $2$-factor with at most $k$ cycles. The paper presents this as a conjecture on strengthening a 2-factor existence theorem to control the number of cycles; the source does not specify its resolution status.