  The excitation energy of deformed intruder states (specifically the 2p2h
bandhead) as a function of proton number $Z$ along $N=20$ is of interest both
in terms of better understanding the evolution of nuclear structure between
spherical $^{40}$Ca and the Island of Inversion nuclei, and for benchmarking
theoretical descriptions in this region. At the center of the $N=20$ Island of
Inversion, the npnh (where n=2,4,6) neutron excitations across a diminished
$N=20$ gap result in deformed and collective ground states, as observed in
$^{32}$Mg. In heavier isotones, npnh excitations do not dominate in the ground
states, but are present in the relatively low-lying level schemes. With the aim
of identifying the expected 2p2h$\otimes\mathrm{s}_{1/2^+}$ state in $^{35}$P,
the only $N=20$ isotone for which the neutron 2p2h excitation bandhead has not
yet been identified, the $^{36}$S(d,$^3$He)$^{35}$P reaction has been revisited
in inverse kinematics with the HELical Orbit Spectrometer (HELIOS) at the
Argonne Tandem Linac Accelerator System (ATLAS). While a candidate state has
not been located, an upper limit for the transfer reaction cross-section to
populate such a configuration within a 2.5 to 3.6\,MeV energy range, provides a
stringent constraint on the wavefunction compositions in both $^{36}$S and
$^{35}$P.
