The induced-path conjecture for the graphs $G_{q,p}$

Let $q\geq 3$ be odd, let $p\geq 1$, and let $G_{q,p}$ be the graph constructed in the paper. Here, $P_t$ denotes the path on $t$ vertices, and a graph is $P_t$-free if it has no induced subgraph isomorphic to $P_t$. Induced-path conjecture. For every $p\geq 1$, the graph $G_{q,p}$ is $P_{3q-2}$-free. This would improve the general bound from Theorem~; the case $q=3$ is supported by the known $P_7$-freeness of $G_{3,p}$, while the conjecture remains open for the other cases discussed here.