The order conjecture for the generalized Lagrangian germ

For coprime positive integers $m,n$, let $W_{m,n}$ be the geometric model obtained by attaching a disk to the one-component link of the singularity $x^m=y^n$, let $Z_{m,n}$ be its Lagrangian submanifold, and let $\phi_{m,n}$ be the symplectic germ obtained from the flow $(x,y)\mapsto(e^{nit}x,e^{mit}y)$ at $t=\frac{2\pi}{m+n}$. Write $\operatorname{Germ}(Z_{m,n},W_{m,n},W_{m,n})$ for the group of germs and $\operatorname{LagGerm}(Z_{m,n},W_{m,n},W_{m,n})$ for the corresponding group of Lagrangian germs. Order conjecture. The germ $\phi_{m,n}$ is trivial in $\operatorname{Germ}(Z_{m,n},W_{m,n},W_{m,n})$ but has order $m+n$ in $\operatorname{LagGerm}(Z_{m,n},W_{m,n},W_{m,n})$. The source presents this as a conjecture in the generalized construction; no resolution is given.