  Theoretical models predict that z~6 quasars are hosted in the most massive
halos of the underlying dark matter distribution and thus would be immersed in
protoclusters of galaxies. However, observations report inconclusive results.
We investigate the 1.1 pMpc2 environment of the z = 7.54 luminous quasar ULAS
J1342+0928. We search for Lyman-break galaxy candidates (LBG) using deep
imaging from the Hubble Space Telescope (HST) in the ACS/F814W,
WFC3/F105W/F125W bands, and Spitzer/IRAC at 3.6 $\mu$m and 4.5 $\mu$m. We
report a zphot = $7.69^{+0.33}_{-0.23}$ LBG with magF125W = 26.41 at 223
projected-pkpc from the quasar. We find no HST counterpart to one [CII]-emitter
previously found with ALMA at 27 projected-pkpc and $z[CII]=7.5341\pm0.0009$
(Venemans et al. 2020). We estimate the completeness of our LBG candidates
using results from CANDELS/GOODS deep blank field searches sharing a similar
filter setup. We find that >50% of the z~7.5 LBGs with magF125W >25.5 are
missed due to the absence of a filter redward of the Lyman-break in F105W,
hindering the UV color accuracy of the candidates. We conduct a QSO-LBG
clustering analysis revealing a low LBG excess of $0.46^{+1.52}_{-0.08}$ in
this quasar field, consistent with an average or low-density field.
Consequently, this result does not present strong evidence of an LBG
overdensity around ULAS J1342+0928. Furthermore, we identify two LBG candidates
with a zphot matching a confirmed z=6.84 absorber along the line-of-sight to
the quasar. All these galaxy candidates are excellent targets for follow-up
observations with JWST and/or ALMA to confirm their redshift and physical
properties.
