Cepheid Metallicity in the Leavitt Law (C-MetaLL) survey: IX. Metallicity dependence of Period-Wesenheit relations based on a homogeneous spectroscopic sample

The C-MetaLL project has provided homogeneous spectroscopic abundances of 290 Classical Cepheids (DCEPs) for which we have the intensity-averaged magnitudes in multiple optical and near-infrared (NIR) bands, periods, pulsation modes, and Gaia parallaxes. Our goal is to derive updated period-Wesenheit-metallicity (PWZ) relations using the largest and most homogeneous metallicity sample ever used for such analyses, covering a range of $-1.3<$[Fe/H]$<+0.3$ dex. We computed several optical and NIR Wesenheit magnitudes using 275 DCEPs with reliable parallaxes, by applying a robust photometric parallax technique, which simultaneously fits all parameters -- including the global Gaia parallax counter-correction -- and handles outliers without data rejection. We find a stronger metallicity dependence ($\gamma \approx -0.5$ mag/dex in optical, $-0.4$ mag/dex in NIR) than recent literature reports. Gaia parallax zero-point conter-corrections ($\epsilon$) vary smoothly across bands, with an average value of $\sim$10 $\mu$as, aligning with previous determinations. Applying our PWZ relations to LMC Cepheids yields distances generally consistent within $1\sigma$ with geometric estimates. The choice of reddening law has a negligible impact, while using only fundamental-mode pulsators significantly increases the uncertainties. Including $\alpha$-element corrections increases $|\gamma|$ and reduces $\epsilon$. However, we find statistically consistent $\gamma$ values with the literature, particularly for the key Wesenheit magnitude in the HST bands, by restricting the sample to the brighter (i.e. closer) objects, or by including only pulsators with $-0.7<$[Fe/H]$<$0.2 dex. Our results hint at a large $\gamma$ or a non-linear dependence on metallicity of DCEP luminosities at the metal-poor end, which is difficult to quantify with the precision of parallaxes of the present dataset.