No perfect state transfer across the longer-path bridge

Let $Z$ be the graph shown in Figure 6: a longer path $X$ joins the relevant subgraphs, with vertices $a,b,c,d$ labelled as in that figure. No-perfect-state-transfer conjecture. Perfect state transfer does not occur between $a$ and $b$ in $Z$.

The authors state that they believe the analogous result holds when the bridging graph is a longer path, but explain that their attempted inductive proof does not succeed. They also note that strong cospectrality transfers from $a,b$ to $c,d$, while perfect state transfer need not, because $c,d$ may have additional eigenvalues in their support.