The Lya emitter (LAE) fraction, X_LAE, is a potentially powerful probe of the evolution of the intergalactic neutral hydrogen gas fraction. However, uncertainties in the measurement of X_LAE are still debated. Thanks to deep data obtained with MUSE, we can measure the evolution of X_LAE homogeneously over a wide redshift range of z~3-6 for UV-faint galaxies (down to M_1500~-17.75). This is significantly fainter than in former studies, and allows us to probe the bulk of the population of high-z star-forming galaxies. We construct a UV-complete photo-redshift sample following UV luminosity functions and measure the Lya emission with MUSE using the second data release from the MUSE HUDF Survey. We derive the redshift evolution of X_LAE for M_1500 in [-21.75;-17.75] for the first time with a equivalent width range EW(Lya)>=65 A and find low values of X_ LAE<~30% at z<~6. For M_1500 in [-20.25;-18.75] and EW(Lya)<~25 A, our X_LAE values are consistent with those in the literature within 1sigma at z<~5, but our median values are systematically lower than reported values over the whole redshift range. In addition, we do not find a significant dependence of X_LAE on M_1500 for EW(Lya)>~50 A at z~3-4, in contrast with previous work. The differences in X_LAE mainly arise from selection biases for Lyman Break Galaxies (LBGs) in the literature: UV-faint LBGs are more easily selected if they have strong Lya emission, hence X_LAE is biased towards higher values. Our results suggest either a lower increase of X_LAE towards z~6 than previously suggested, or even a turnover of X_LAE at z~5.5, which may be the signature of a late or patchy reionization process. We compared our results with predictions from a cosmological galaxy evolution model. We find that a model with a bursty star formation (SF) can reproduce our observed X_LAE much better than models where SF is a smooth function of time.
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