  We study the molecular gas content of 24 star-forming galaxies at $z=3-4$,
with a median stellar mass of $10^{9.1}$ M$_{\odot}$, from the MUSE Hubble
Ultra Deep Field (HUDF) Survey. Selected by their Lyman-alpha-emission and
H-band magnitude, the galaxies show an average EW $\approx 20$ angstrom, below
the typical selection threshold for Lyman Alpha Emitters (EW $> 25$ angstrom),
and a rest-frame UV spectrum similar to Lyman Break Galaxies. We use rest-frame
optical spectroscopy from KMOS and MOSFIRE, and the UV features observed with
MUSE, to determine the systemic redshifts, which are offset from Lyman alpha by
346 km s$^{-1}$, with a 100 to 600 km s$^{-1}$ range. Stacking CO(4-3) and
[CI](1-0) (and higher-$J$ CO lines) from the ALMA Spectroscopic Survey of the
HUDF (ASPECS), we determine $3\sigma$ upper limits on the line luminosities of
$4.0\times10^{8}$ K km s$^{-1}$pc$^{2}$ and $5.6\times10^{8}$ K km
s$^{-1}$pc$^{2}$, respectively (for a 300 km s$^{-1}$ linewidth). Stacking the
1.2 mm and 3 mm dust continuum flux densities, we find a $3\sigma$ upper limits
of 9 $\mu$Jy and $1.2$ $\mu$Jy, respectively. The inferred gas fractions, under
the assumption of a 'Galactic' CO-to-H$_{2}$ conversion factor and gas-to-dust
ratio, are in tension with previously determined scaling relations. This
implies a substantially higher $\alpha_{\rm CO} \ge 10$ and $\delta_{\rm GDR}
\ge 1200$, consistent with the sub-solar metallicity estimated for these
galaxies ($12 + \log(O/H) \approx 7.8 \pm 0.2$). The low metallicity of $z \ge
3$ star-forming galaxies may thus make it very challenging to unveil their cold
gas through CO or dust emission, warranting further exploration of alternative
tracers, such as [CII].
