The ionizing properties of two bright Ly$\alpha$ emitters in the BDF reionized bubble at z=7

We investigate the ionizing properties of the pair of bright Ly$\alpha$ emitting galaxies BDF521 and BDF2195 at z=7.012 in order to constrain their contribution to the formation of the BDF "reionized bubble". We obtain constraints on UV emission lines (CIV$\lambda 1548$ doublet, HeII$\lambda 1640$, OIII]$\lambda 1660$ doublet, and CIII]$\lambda 1909$ doublet) from deep VLT-XSHOOTER observations and compare them to those available for other high-redshift objects, and to models with mixed stellar and AGN emission. We use this spectroscopic information together with the photometry available in the field to constrain the physical properties of the two objects using the spectro-photometric fitting code BEAGLE. We do not detect any significant emission at the expected position of UV lines, with 3$\sigma$ upper limits of EW$\lesssim$2-7 AA rest-frame. We find that the two objects have lower CIII] emission than expected on the basis of the correlation between the Ly$\alpha$ and CIII] EWs. The EW limits on CIV and HeII emission exclude pure AGN templates at $\sim2-3\sigma$ significance, and only models with a $\lesssim$40% AGN contribution are compatible with the observations. The two objects are found to be relatively young ($\sim$20-30 Myrs) and metal-poor ($\lesssim 0.3 Z_{\odot}$) with stellar masses of a few $10^9M_{\odot}$. Their production rate of hydrogen ionizing photons per intrinsic UV luminosity is log($\xi_{ion}^*$/Hz erg$^{-1}$)=25.02-25.26, consistent with values typically found in high-redshift galaxies, but more than twice lower than values measured in $z>$7 galaxies with strong CIII] and/or optical line emission ($\simeq$25.6-25.7). The two BDF emitters have no evidence of higher than average ionizing capabilities and are not capable of reionizing their surroundings by their own means under realistic assumptions on the escape fraction of ionizing photons. (abridged)