  The recent observations on WASP-39 b by JWST have revealed hints of high
metallicity within the atmosphere compared to its host star (Feinstein et al.
2022; Ahrer et al. 2023; Alderson et al. 2023; Rustamkulov et al. 2023; Tsai et
al. 2023). There are various theories on how these high metallic atmospheres
emerge. In this study, we closely investigate the impact of extreme escape in
the form of hydrodynamic escape to see its impact on atmospheric metallicity
and spectral features such as CH$_4$, CO$_2$, and SO$_2$. We perform a grid
simulation, with an adapted version of MESA that includes hydrodynamic escape
(Kubyshkina et al. 2018; 2020), to fully evolve planets with similar masses and
radii to the currently observed WASP-39 b estimates. By making use of
(photo-)chemical kinetics and radiative transfer codes, we evaluate the
transmission spectra at various time intervals throughout the simulation. Our
results indicate that the massive size of WASP-39 b limits the metal
enhancement to a maximum of ~1.23 the initial metallicity. When incorporating
metal drag, this enhancement factor is repressed to an even greater degree,
resulting in an enrichment of at most ~0.4%. As a consequence, when assuming an
initial solar metallicity, metal-enriched spectral features like SO$_2$ are
still missing after ~9 Gyr into the simulation. This paper, thus, demonstrates
that hydrodynamic escape cannot be the primary process behind the high
metallicity observed in the atmosphere of WASP-39 b, suggesting instead that a
metal-enhanced atmosphere was established during its formation.
