  We present the results obtained from follow-up observations of the MUSE Ultra
Deep Field (MUDF) at X-ray energies with XMM-Newton. The MUDF is centred on a
unique field with two bright, physically associated quasars at $z\simeq3.23$,
separated by $\sim$500 kpc in projection. Both quasars are embedded within
extended Ly$\alpha$ nebulae ($\gtrsim 100~\rm kpc$ at a surface brightness flux
level of $\approx 6\times 10^{-19} \rm erg~s^{-1}~cm^{-2}~arcsec^{-2}$), whose
elongated morphology is suggestive of an extended filament connecting the
quasar haloes. The new X-ray observations presented here allow us to
characterise the physical properties (e.g. X-ray slope, luminosities, gas
column densities) in the innermost region of the MUDF quasars. We find that
both quasars are X-ray underluminous compared to objects at similar ultraviolet
luminosities. Based on our X-ray spectral analysis, absorbing columns of
$N_H(z)\gtrsim$ 10$^{23}$ cm$^{-2}$ appear unlikely, therefore such a weakness
is possibly intrinsic. When also including literature data, we do not observe
any detectable trend between the area of the nebulae and nuclear luminosities
at both the rest-frame 2 keV and 2500 $\rm \mathring{A}$. The area is also not
correlated with the X-ray photon index nor with the integrated band flux in the
hard band (2$-$10 keV). We also do not find any trend between the extended
Ly$\alpha$ emission of the nebulae and the nuclear X-ray luminosity. Finally,
the properties of the MUDF quasars' nebulae are consistent with the observed
relation between the Ly$\alpha$ integrated luminosity of the nebulae and their
area. Our results suggest that the quasar ionization power is not a strong
driver of the morphology and size of the nebulae.

Astrophysics of Galaxies;High Energy Astrophysical Phenomena