A critical analysis of main-sequence fitting in open clusters to derive the helium-to-metal enrichment ratio $\Delta Y/\Delta Z$
We aim to investigate the feasibility of accurately determining the helium-to-metal enrichment ratio $\Delta Y/\Delta Z$ for open clusters using Gaia DR3 photometry. To test the reliability of this calibration, we performed a theoretical investigation using mock open clusters. We generated synthetic photometric data from isochrones calculated by five different stellar evolution codes (FRANEC, PARSEC 1.2s, PARSEC 2.0, BASTI, and MIST), for which the true $\Delta Y/\Delta Z$ is known. We then fitted these mock clusters with two sets of isochrones calculated with the FRANEC code, differing only in the implementation of bolometric corrections (BCs). The analysis focused on the G-band absolute magnitude range (4.3 to 6.5 mag) to minimise the impact of poorly constrained physics. Synthetic clusters were generated at [Fe/H] values from 0.0 to 0.15 dex, for different numbers of populating stars and different levels of photometric uncertainties. The Monte Carlo experiments revealed significant and code-dependent biases. Unbiased results were achieved only when the stellar models used for synthetic-cluster generation and fitting were identical. Using identical FRANEC stellar models but different BCs introduced a significant bias of up to 0.6. Furthermore, using different stellar models for synthetic cluster generations resulted in even larger biases: $\Delta Y/\Delta Z$ was underestimated by up to 0.8 for PARSEC target isochrones, while it was overestimated for BASTI and MIST isochrones by up to 0.6 and 1.5, respectively. The magnitude and the inconsistency of these biases strongly suggest that the photometric calibration of $\Delta Y/\Delta Z$ using open clusters is not reliably robust.