  One-neutron removal $(p,pn)$ reactions induced by two-neutron Borromean
nuclei are studied within a Transfer-to-the-Continuum (TC) reaction framework,
which incorporates the three-body character of the incident nucleus. The
relative energy distribution of the residual unbound two-body subsystem, which
is assumed to retain information on the structure of the original three-body
projectile, is computed by evaluating the transition amplitude for different
neutron-core final states in the continuum. These transition amplitudes depend
on the overlaps between the original three-body ground-state wave function and
the two-body continuum states populated in the reaction, thus ensuring a
consistent description of the incident and final nuclei. By comparing different
$^{11}$Li three-body models, it is found that the $^{11}$Li$(p,pn){^{10}}$Li
relative energy spectrum is very sensitive to the position of the $p_{1/2}$ and
$s_{1/2}$ states in $^{10}$Li and to the partial wave content of these
configurations within the $^{11}$Li ground-state wave function. The possible
presence of a low-lying $d_{5/2}$ resonance is discussed. The coupling of the
single particle configurations with the non-zero spin of the $^{9}$Li core,
which produces a spin-spin splitting of the states, is also studied. Among the
considered models, the best agreement with the available data is obtained with
a $^{11}$Li model that incorporates the actual spin of the core and contains
$\sim$31\% of $p_{1/2}$-wave content in the $n$-$^9$Li subsystem, in accord
with our previous findings for the $^{11}$Li(p,d)$^{10}$Li transfer reaction,
and a near-threshold virtual state.
