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Recent work has suggested extreme [OIII] emitting star-forming galaxies areimportant to reionization. Relatedly, [OIII]/[OII] has been put forward as anindirect estimator of the Lyman Continuum (LyC) escape fraction ($f_{esc}$) at$z\gtrsim4.5$ when the opaque IGM renders LyC photons unobservable. Using deeparchival 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($f_{esc}^{rel} < 6.3^{+0.7}_{-0.7} \%$ at $1\sigma$) in a deep U-band stack ofour sample (31.98 mag at 1$\sigma$). This constraint is in agreement withrecent studies of star-forming galaxies spanning $z\sim1-4$ that have foundvery low average $f_{esc}$. Despite the galaxies in our study having anestimated average rest-frame EW([OIII]$\lambda5007$)$\sim400\AA$ and[OIII]/[OII]$\sim 4$ from composite SED-fitting, we find no LyC detection,which brings into question the potential of [OIII]/[OII] as an effective probeof the LyC--a majority of LyC emitters have [OIII]/[OII]$>3$, but we establishhere that [OIII]/[OII]$>3$ does not guarantee significant LyC leakage for apopulation. Since even extreme star-forming galaxies are unable to produce the$f_{esc}\sim10-15\%$ required by most theoretical calculations for star-forminggalaxies to drive reionization, there must either be a rapid evolution of$f_{esc}$ between $z\sim3.5$ and the Epoch of Reionization, or hithertoobservationally unstudied sources (e.g. ultra-faint low-mass galaxies with$\log(M/M_\odot)\sim7-8.5$) must make an outsized contribution to reionization.

A Low Lyman Continuum Escape Fraction of $<10\%$ for Extreme [OIII] Emitters in an Overdensity at z$\sim3.5$