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No strong dependence of Lyman continuum leakage on physical properties of star-forming galaxies at $\mathbf{3.1 \lesssim z \lesssim 3.5}$

We present Lyman continuum (LyC) radiation escape fraction $f_{\rm{esc}}$ measurements for 183 spectroscopically confirmed star-forming galaxies in the redshift range $3.11 < z < 3.53$ in the \textit{Chandra} Deep Field South. We use ground-based imaging to measure $f_{\rm{esc}}$, and use ground- and space-based photometry to derive galaxy physical properties using spectral energy distribution (SED) fitting. We additionally derive [O III]+H$β$ equivalent widths (that fall in the observed K band) by including nebular emission in the SED fitting. After removing foreground contaminants, we report the discovery of 11 new candidate LyC leakers, with absolute LyC escape fractions, $f_{\rm{esc}}$ in the range $0.14-0.85$. From non-detections, we place $1σ$ upper limits of $f_{\rm{esc}}<0.12$, where the Lyman-break selected galaxies have $f_{\rm{esc}} < 0.11$ and `blindly' discovered galaxies with no prior photometric selection have $f_{\rm{esc}}<0.13$. We find a slightly higher $1σ$ limit of $f_{\rm{esc}}<0.20$ for extreme emission line galaxies with rest-frame [O III]+H$β$ equivalent widths $>300$A. For candidate LyC leakers, we find a weak negative correlation between $f_{\rm{esc}}$ and galaxy stellar masses, no correlation between $f_{\rm{esc}}$ specific star-formation rates (sSFRs) and a positive correlation between $f_{\rm{esc}}$ and EW$_0$([O III]+H$β$). The weak/no correlations between stellar mass and sSFRs may be explained by misaligned viewing angles and/or non-coincident timescales of starburst activity and periods of high $f_{\rm{esc}}$. Alternatively, escaping radiation may predominantly occur in highly localised star-forming regions, or $f_{\rm{esc}}$ measurements may be impacted by stochasticity of the intervening neutral medium, obscuring any global trends with galaxy properties. These hypotheses have important consequences for models of reionisation.