A low Lyman Continuum escape fraction of <10 per cent for extreme [O iii] emitters in an overdensity at z ∼ 3.5

Recent work has suggested extreme [OIII] emitting star-forming galaxies are important to reionization. Relatedly, [OIII]/[OII] has been put forward as an indirect estimator of the Lyman Continuum (LyC) escape fraction ($f_{esc}$) at $z\gtrsim4.5$ when the opaque IGM renders LyC photons unobservable. Using deep archival U-band (VLT/VIMOS) imaging of a recently confirmed overdensity at $z\sim3.5$ we calculate tight constraints on $f_{esc}$ for a sample (N=73) dominated by extreme [OIII] emitters. We find no Lyman Continuum signal ($f_{esc}^{rel} 3$, but we establish here that [OIII]/[OII]$>3$ does not guarantee significant LyC leakage for a population. Since even extreme star-forming galaxies are unable to produce the $f_{esc}\sim10-15\%$ required by most theoretical calculations for star-forming galaxies to drive reionization, there must either be a rapid evolution of $f_{esc}$ between $z\sim3.5$ and the Epoch of Reionization, or hitherto observationally unstudied sources (e.g. ultra-faint low-mass galaxies with $\log(M/M_\odot)\sim7-8.5$) must make an outsized contribution to reionization.