Dallard, Milani\v{c}, and \v{S}torgel, together with an independent conjecture by Gartland and Lokshtanov, claimed that for any planar graph $H$, both Maximum Weight Independent Set and List $3$‑Coloring are polynomial‑time solvable on $H$‑induced‑minor‑free graphs. Gartland and Lokshtanov further hypothesized that such graphs admit balanced separators dominated by a constant number of vertices as a structural explanation. Bonnet and Chang recently refuted the independent‑set part by showing that Maximum Independent Set remains $\mathsf{NP}$‑hard even when the $5\times5$ grid is excluded as an induced minor. We revisit both conjectures. First, the hard instances of Bonnet and Chang actually exclude much simpler planar graphs, including several $16$‑vertex trees. We introduce a second, independent family of hard instances and prove that $3$‑Coloring is also $\mathsf{NP}$‑hard in both settings, thereby fully refuting the algorithmic conjecture, even in its quasipolynomial form unless $\mathsf{NP}\subseteq\mathsf{QP}$. On the structural side, we identify planar graphs $H$ for which the separator conjecture holds while the algorithmic one fails, showing that constantly dominated balanced separators alone cannot account for the tractability of Maximum Weight Independent Set and List $3$‑Coloring.
Review: The paper delivers two new constructions that decisively break the polynomial‑time solvability conjectures for maximum independent set and list $3$‑coloring in planar induced‑minor‑free graphs, and demonstrates that balanced separators are not a sufficient structural reason for tractability, guiding future research directions.