The polarized cross section ratio $\sigma_{LT'}/\sigma_{0}$ from hard exclusive $\pi^{-} \Delta^{++}$ electroproduction off an unpolarized hydrogen target has been extracted based on beam-spin asymmetry measurements using a 10.2 GeV / 10.6 GeV incident electron beam and the CLAS12 spectrometer at Jefferson Lab. The study, which provides the first observation of this channel in the deep-inelastic regime, focuses on very forward-pion kinematics in the valence regime, and photon virtualities ranging from 1.5 GeV$^{2}$ up to 7 GeV$^{2}$. The reaction provides a novel access to the $d$-quark content of the nucleon and to $p \rightarrow \Delta^{++}$ transition generalized parton distributions. A comparison to existing results for hard exclusive $\pi^{+} n$ and $\pi^{0} p$ electroproduction is provided, which shows a clear impact of the excitation mechanism, encoded in transition generalized parton distributions, on the asymmetry.
