The non-occurrence conjecture for PGST in double blow-ups of paths

Let $P_n$ be the path on $n$ vertices, and let $\overset{2}{\uplus}~P_n$ denote its double blow-up, with corresponding vertices $(0,u)$ and $(1,u)$ for each vertex $u$ of $P_n$. Let

$$n=2^t r-1,$$

where $t\geq 2$ and $r$ is an odd prime number. If $u$ is a multiple of $2^{t-1}$, then the non-occurrence conjecture states that PGST does not occur between $(0,u)$ and $(1,u)$ in $\overset{2}{\uplus}~P_n$. This conjecture extends the demonstrated obstruction for $P_{11}$; its general validity is not established in the supplied text.