C. AccardM. BétherminM. BoquienV. BuatL. ValliniF. RenaudK. KraljicM. AravenaP. CassataE. da CunhaP. DamI. de LoozeM. Dessauges-ZavadskyY. DuboisA. FaisstY. FudamotoM. GinolfiC. GruppioniS. HanR. Herrera-CamusH. InamiA. M. KoekemoerB. C. LemauxJ. LiY. LiB. MobasherJ. MolinaA. NanniM. PallaF. PozziM. RelañoM. RomanoP. SawantJ. SpilkerA. TsujitaE. VeraldiV. VillanuevaW. WangS. K. YiG. Zamorani2026-02-252026-02-252025-1010.1051/0004-6361/202556140https://cris-uv-2.scimago.es/handle/123456789/8510<jats:p> <jats:italic>Context.</jats:italic> Star formation governs galaxy evolution, shaping stellar mass assembly and gas consumption across cosmic time. The Kennicutt-Schmidt (KS) relation, linking the star formation rate (SFR) and gas surface densities, is fundamental to understand star formation regulation, yet remains poorly constrained at <jats:italic>z</jats:italic>  &gt; 2 due to observational limitations and uncertainties in locally calibrated gas tracers. The [CII] 158 μm line has recently emerged as a key probe of the cold ISM and star formation in the early Universe. </jats:p> <jats:p> <jats:italic>Aims.</jats:italic> We investigate whether the resolved [CII]–SFR and KS relations established at low redshift remain valid at 4 &lt;  <jats:italic>z</jats:italic>  &lt; 6 by analysing 13 main-sequence galaxies from the ALPINE and CRISTAL surveys, using multi-wavelength data (HST, JWST, ALMA) at ∼2 kpc resolution. </jats:p> <jats:p> <jats:italic>Methods.</jats:italic> We performed pixel-by-pixel spectral energy distribution (SED) modelling with CIGALE on resolution-homogenised images. We developed a statistical framework to fit the [CII]–SFR relation that accounts for pixel covariance and compare our results to classical fitting methods. We tested two [CII]-to-gas conversion prescriptions to assess their impact on inferred gas surface densities and depletion times. </jats:p> <jats:p> <jats:italic>Results.</jats:italic> We find a resolved [CII]–SFR relation with a slope of 0.87 ± 0.15 and intrinsic scatter of 0.19 ± 0.03 dex, which is shallower and tighter than previous studies at <jats:italic>z</jats:italic>  ∼ 5. The resolved KS relation is highly sensitive to the [CII]-to-gas conversion factor: using a fixed global <jats:italic>α</jats:italic> <jats:sub>[CII]</jats:sub> yields depletion times of 0.5–1 Gyr, while a surface brightness-dependent <jats:italic>W</jats:italic> <jats:sub>[CII]</jats:sub> , accounting for local ISM conditions, places some galaxies with high gas density in the starburst regime (&lt; 0.1 Gyr). </jats:p> <jats:p> <jats:italic>Conclusions.</jats:italic> Future inputs from both simulations and observations are required to better understand how the [CII]-to-gas conversion factor depends on local ISM properties. We need to break this fundamental limit to properly study the KS relation at <jats:italic>z</jats:italic>  ≳ 4. </jats:p>journal-article