degrees of freedom are decoupled and the quantum number of the transversal $2D$ HO modes can be used to define the asymptotic scattering channel. $$\psi_n(r)=e^{iq_{n}z}{\phi_{n}(\rho)}+\sum_{n^{\prime}=0}^{n_o}f_{nn^{\prime}}^{\pm}e^{iq_{n^\prime}|z|}{\phi_{n^{\prime}}(\rho)},
\label{eq:1d_scattering_wavefunction}$$ describing an incoming wave in channel $n$ which is then (in)-elastically scattered into all open channels. Here, the scattering amplitude $f_{nn^{\prime}}^{\pm}$ in forward $(+)$, respectively backward $(-)$ direction reads $$f_{nn^{\prime}}^{\pm}=f_{nn^{\prime}}^e+\operatorname{sgn}(z)f_{nn^{\prime}}^o,
\label{eq:scattering_forward_backward}$$ whereas in turn $f^{e}$ and $f^{o}$ refer to the respective scattering amplitudes for even and odd exchange symmetry and, $\operatorname{sgn}(z)=z/|z|$ denotes the sign function. By the conservation of flux, the forward $f^+$ and backward $f^-$ scattering amplitude contain the same information about a scattering event, we concentrate our analysis on $f^{+}$, for which the transmission and reflection coefficients $T_{{n{n^{\prime}}}}$ and $R_{{n{n^{\prime}}}}$, respectively, from channel $n$ to ${n^{\prime}}$ take the following form $$\begin{aligned}
  T_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})} &= |\delta_{{n{n^{\prime}}}}+f^{\ell}_{{n{n^{\prime}}}}+f^{{\ell^{\prime}}}_{{n{n^{\prime}}}}|^2
\label{eq:transmission_fplus}
\\
R_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})} &= |f^{\ell}_{{n{n^{\prime}}}}+f^{{\ell^{\prime}}}_{{n{n^{\prime}}}}|^2,
\label{eq:reflection_fplus}\end{aligned}$$ where $\ell$ and ${\ell^{\prime}}$ refer to even and odd partial waves, respectively. The transition probability $W_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})}$ characterizing the transversal excitation and de-excitation processes from channel $n$ into a specific channel ${n^{\prime}}$ are given by the sum of the corresponding transmission and reflection coefficients $$W_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})}=T_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})}+R_{{n{n^{\prime}}}}^{(\ell,{\ell^{\prime}})}
\label{eq:transition_nnp}$$ If the constituents of the scattering event both belong to the same symmetry class, i.e. both are 