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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 HubbleUltra Deep Field (HUDF) Survey. Selected by their Lyman-alpha-emission andH-band magnitude, the galaxies show an average EW $\approx 20$ angstrom, belowthe typical selection threshold for Lyman Alpha Emitters (EW $> 25$ angstrom),and a rest-frame UV spectrum similar to Lyman Break Galaxies. We use rest-frameoptical spectroscopy from KMOS and MOSFIRE, and the UV features observed withMUSE, to determine the systemic redshifts, which are offset from Lyman alpha by346 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 theHUDF (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 kms$^{-1}$pc$^{2}$, respectively (for a 300 km s$^{-1}$ linewidth). Stacking the1.2 mm and 3 mm dust continuum flux densities, we find a $3\sigma$ upper limitsof 9 $\mu$Jy and $1.2$ $\mu$Jy, respectively. The inferred gas fractions, underthe assumption of a 'Galactic' CO-to-H$_{2}$ conversion factor and gas-to-dustratio, are in tension with previously determined scaling relations. Thisimplies a substantially higher $\alpha_{\rm CO} \ge 10$ and $\delta_{\rm GDR}\ge 1200$, consistent with the sub-solar metallicity estimated for thesegalaxies ($12 + \log(O/H) \approx 7.8 \pm 0.2$). The low metallicity of $z \ge3$ star-forming galaxies may thus make it very challenging to unveil their coldgas through CO or dust emission, warranting further exploration of alternativetracers, such as [CII].

Measuring the average molecular gas content of star-forming galaxies at $z=3-4$