A Low Lyman Continuum Escape Fraction of $<10\%$ for Extreme [OIII]
  Emitters in an Overdensity at z$\sim3.5$
  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
($f_{esc}^{rel} < 6.3^{+0.7}_{-0.7} \%$ at $1\sigma$) in a deep U-band stack of
our sample (31.98 mag at 1$\sigma$). This constraint is in agreement with
recent studies of star-forming galaxies spanning $z\sim1-4$ that have found
very low average $f_{esc}$. Despite the galaxies in our study having an
estimated 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 probe
of the LyC--a majority of LyC emitters have [OIII]/[OII]$>3$, 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.
