Order bounds for 2K2-split minimal (∞, k)-polar obstructions

Let $k$ be an integer with $k\ge 3$, and let $G=(C,S,I)$ be a $2K_2$-split minimal $(\infty,k)$-polar obstruction. Write $c=|C|$ and $i=|I|$ for the numbers of vertices in $C$ and $I$, respectively.

Order-bound conjecture. Then

$$k\le i\le 2k-2\quad\text{and}\quad c\le 2k-i-1.$$

This conjecture gives sharper structural bounds on the clique and independent-set parts of minimal $(\infty,k)$-polar obstructions and could help establish improved bounds on their order. The source presents it as an initial conjecture; no resolution is supplied.