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In this study, we estimate the fraction of binaries with high mass ratios for 202 open clusters in the extended solar neighbourhood (closer than 1.5 kpc from the Sun). This is one of the largest homogeneous catalogues of multiplicity fractions in open clusters to date, including the unresolved and total (close-binary) multiplicity fractions of main-sequence systems with mass ratio larger than $0.6_{-0.15}^{+0.05}$. The unresolved multiplicity fractions are estimated applying a flexible mixture model to the observed Gaia colour-magnitude diagrams of the open clusters. Then we use custom Gaia simulations to account for the resolved systems and derive the total multiplicity fractions. The studied open clusters have ages between 6.6 Myr and 3.0 Gyr and total high-mass-ratio multiplicity fractions between 6% and 80%, with a median of 18%. The multiplicity fractions increase with the mass of the primary star, as expected. The average multiplicity fraction per cluster displays an overall decreasing trend with the open cluster age up to ages about 100 Myr, above which the trend increases. Our simulations show that most of this trend is caused by complex selection effects (introduced by the mass dependence of the multiplicity fraction and the magnitude limit of our sample). Furthermore, the multiplicity fraction is not significantly correlated with the clusters' position in the Galaxy. The spread in multiplicity fraction decreases significantly with the number of cluster members (used as a proxy for cluster mass). We also find that the multiplicity fraction decreases with metallicity, in line with recent studies using field stars., Comment: 17 pages, 13 figures, accepted for publication in A&A after minor modifications