Schiffler Problem 6.3

For admissible parameters $a,b$ and any two lattice paths $P,Q\in\mathcal{D}(a,b)$, does equality of their Lagrange numbers imply that their associated band graphs are isomorphic? That is, is $\operatorname{Lag}(P)=\operatorname{Lag}(Q)$ sufficient to conclude $\operatorname{Band}(P)\cong\operatorname{Band}(Q)$? This is false: there exist $P,Q\in\mathcal{D}(17,9)$ such that $\operatorname{Lag}(P)=\operatorname{Lag}(Q)$ but $\operatorname{Band}(P)\not\cong\operatorname{Band}(Q)$.