  Contrary to expectations from scenarios of black hole growth driven by galaxy
interactions and mergers, dual active galactic nuclei (AGN) with kiloparsec
separations are rarely observed and are very difficult to identify, in
particular at high redshifts (i.e. z>2). Focussing on the recently discovered
dual AGN system LBQS 0302-0019 at z=3.29, we seek to identify further group
members in its environment and to understand their formation history through
deep high-angular-resolution imaging. We present deep Hubble Space Telescope
(HST) Wide-field Camera 3 near-infrared imaging of LBQS 0302-0019. In
combination with ground-based VLT/HAWK-I imaging, we infer accurate sizes,
colours, ages, and stellar masses of companion galaxies. We clearly detect four
companion objects close to LBQS 0302-0019 that also have faint signatures in
the ground-based images. We constrain light-weighted ages and masses for the
two most prominent companions, Jil1 and Jil2, to $t_\star=252_{-109}^{+222}$Myr
with $\log(M_\star/[\mathrm{M}_\odot])= 11.2_{-0.1}^{+0.3}$ and
$t_{\star}=19_{-14}^{+74}$Myr with $\log(M_\star/[\mathrm{M}_\odot])=
9.4_{-0.4}^{+0.9}$, respectively. The HST data also show that the obscured AGN,
previously identified by strong nebular HeII emission, is associated with the
young massive companion Jil2. Because very massive stars of the starburst
cannot be solely responsible for the HeII emission, we strengthen our initial
conclusion that Jil2 has been hosting an AGN. If the young starburst of Jil2
had been accompanied by sustained black hole growth, Jil2 may have contributed
HeII-ionising flux to create the large HeII Ly$\alpha$ proximity zone around
LBQS 0302-0019. Hence, the duration of the current luminous AGN episode of LBQS
0302-0019 may have been overestimated.
