The generalized $k$-independence lower-bound conjecture

Let $k\geq 2$ be an integer, and let $R_i$ be the sequence of trees of order $ki$ defined by taking $R_1$ to be a $k$-tree and, for $i\geq 2$, obtaining $R_i$ by adding an edge joining a vertex of a member of $R_{i-1}$ to a vertex of a $k$-tree. For a simple graph $G$, let $\alpha_k(G)$ denote its generalized $k$-independence number, let $n=|V(G)|$, let $\omega(G)$ be the dimension of its cycle space, and let $\Gamma_G$ be the graph whose components occur in the claimed characterization.

Generalized $k$-independence conjecture. For every $n$-vertex simple graph $G$,

$$\alpha_k(G)\geq \frac{k-1}{k}[n-\omega(G)].$$

Equality holds if and only if all the following conditions hold: the cycles of $G$, if any, are pairwise vertex-disjoint; every cycle of $G$, if any, has order congruent to $1$ modulo $k$; and each component $T$ of $\Gamma_G$ satisfies $|V_T|\equiv 0\pmod{k}$ and belongs to $R_{|V_T|/k}$. This conjecture extends the paper's proved sharp bound for $k=4$ to general integers $k\geq 2$; the proposed equality characterization depends on the recursively defined trees $R_i$.