The GI-hardness conjecture for complementary subgraph partitioning on long-hole-free graphs

Let $C_{k\geq 6}$-free graphs be graphs containing no induced cycle of length at least $6$. The problem $\textsc{Comp-Sub}(\mathcal{PM})$ asks whether a graph has a complementary decomposition whose edge cut is a perfect matching.

GI-hardness conjecture. $\textsc{Comp-Sub}(\mathcal{PM})$ is $\GI$-hard on $C_{k\geq 6}$-free graphs.

The conjecture concerns the unresolved complexity gap for graphs with no induced cycle of length at least $6$; the source establishes related results for other values of the forbidden-cycle threshold but leaves the case $k=6$ open.