  Recently, it was noted that absorptive corrections to the electromagnetic
form factor in high-energy proton-proton scattering are important for the
theoretical interpretation of the $p^\uparrow{p}$ and $p^\uparrow{A}$ analyzing
power $A_\text{N}(t)$ measurements with the Hydrogen Jet Target polarimeter
(HJET) at RHIC. Here, a concise expression for the absorptive correction was
derived within the eikonal approach. The resulting analysis reveals a
systematic bias, nearly independent of the beam energy, in the experimental
determination of the real-to-imaginary ratio $\rho$ when absorption effects are
overlooked in the data analysis. Quantification of this bias, as
$\rho^\text{meas}=\rho + (0.036\pm0.016)_\text{bias}$, was achieved using a
Regge fit applied to available proton-proton measurements of
$\rho^\text{meas}(s)$ and $\sigma^\text{meas}_\text{tot}(s)$. Considering the
potential impact of such an effect on the experimentally determined
$A_\text{N}(t)$, one may enhance consistency between the HJET and STAR
measurements of the hadronic spin-flip amplitude. While the sign of the bias in
the value of $\rho$ aligns with the anticipated effective increase in the
proton charge radius in $pp$ scattering due to absorption, it amplifies the
observed discrepancy between $\sigma^\text{meas}_\text{tot}$ and
$\rho^\text{meas}$ values at $\sqrt{s}=13\,\text{TeV}$ as measured in the TOTEM
experiment. Evaluation (using published TOTEM data) of the measured
proton-proton $d\sigma/dt$ dependence on the absorptive corrections indicated
that possible soft photon corrections to the hadronic amplitude slope may be
essential for such data analysis.
