Young open clusters (t<200 Myr) have been observed to exhibit several peculiarities in their chemical compositions, from a slightly sub-solar iron content, super-solar abundances of some atomic species (e.g. ionised chromium), and atypical enhancements of [Ba/Fe], with values up to +0.7 dex. Regarding the behaviour of the other $s$-process elements like yttrium, zirconium, lanthanum, and cerium, there is general disagreement in the literature. In this work we expand upon our previous analysis of a sample of five young open clusters (IC2391, IC2602, IC4665, NGC2516, and NGC2547) and one star-forming region (NGC2264), with the aim of determining abundances of different neutron-capture elements, mainly CuI, SrI, SrII, YII, ZrII, BaII, LaII, and CeII. We analysed high-resolution, high signal-to-noise spectra of 23 solar-type stars observed within the textit{Gaia}-ESO survey. We find that our clusters have solar [Cu/Fe] within the uncertainties, while we confirm the super-solar [Ba/Fe] values (from +0.22 to +0.64 dex). Our analysis also points to mildly enhanced [Y/Fe] values (from 0 and +0.3 dex). For the other $s$-process elements we find that [X/Fe] ratios are solar at all ages. It is not possible to reconcile the anomalous behaviour of Ba and Y at young ages with standard stellar yields and Galactic chemical evolution model predictions. Thus, we explore different possible scenarios related to the behaviour of spectral lines, from the sensitivity to the presence of magnetic fields to the first ionisation potential effect. We also investigate the possibility that they may arise from alterations of the structure of the stellar photosphere due to higher levels of activity in such young stars. We are still unable to explain these enhancements, but we suggest that other elements (i.e. La) might be more reliable tracer of the $s$-process at young ages and encourage further observations.
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