Context. A sample of 895 s-process-rich candidates has been found among the 454 180 giant stars surveyed by LAMOST at low spectral resolution (R ~ 1800). In a previous study, taking advantage of the higher resolution (R ~ 86 000) offered by the HERMES-Mercator spectrograph, we performed a reanalysis of 15 of the brightest stars of this sample. Among these 15 programme stars, which have close-to-solar metallicities, 11 showed mild to strong heavy-element overabundances. However, we did not investigate the nucleosynthetic process(es) at the origin of these overabundances in our former study. Aims. We derive the abundances in s- and r-process elements of the 15 targets in order to investigate whether some stars also show an i-process signature, as sometimes found in their lower-metallicity counterparts (namely, the carbon-enhanced metal-poor (CEMP)-rs stars). Methods. Abundances are derived from the high-resolution HERMES spectra for Pr, Nd, Sm, and Eu using the TURBOSPECTRUM radiative-transfer local thermodynamic equilibrium code with MARCS model atmospheres. Results. Using the new classification scheme proposed in our recent study, we find that two stars show overabundances in both sand r-process elements that are well above the level expected from Galactic chemical evolution (GCE), an analogous situation to that of CEMP-rs stars at lower metallicities. We compare the abundances of the most enriched stars with the nucleosynthetic predictions from the STAREVOL stellar evolutionary code and find abundances compatible with an i-process occurring in asymptotic giant branch stars. Conclusions. Despite there being a larger number of heavy elements with which to characterize the enrichment pattern, the limit between CEMP-s and CEMP-rs stars remains indistinct. Nevertheless, it is interesting to note that an increasing number of extrinsic stars are found to have abundances better reproduced by an i-process pattern even at close-to-solar metallicities.