  Both ($e$,$e'p$) and ($p$,$2p$) reactions have been performed to study the
proton single-particle character of nuclear states with its related
spectroscopic factor. Recently, the dispersive optical model (DOM) was applied
to the ($e$,$e'p$) analysis revealing that the traditional treatment of the
single-particle overlap function, distorted waves, and nonlocality must be
further improved to achieve quantitative nuclear spectroscopy. We apply the DOM
wave functions to the traditional ($p$,$2p$) analysis and investigate the
consistency of the DOM spectroscopic factor that describes the ($e$,$e'p$)
cross section with the result of the ($p$,$2p$) analysis. Additionally, we make
a comparison with a phenomenological single-particle wave function and optical
potential. Uncertainty arising from a choice of $p$-$p$ interaction is also
investigated. We implement the DOM wave functions to the distorted wave impulse
approximation (DWIA) framework for ($p$,$2p$) reactions. DOM + DWIA analysis on
$^{40}$Ca($p$,$2p$)$^{39}$K data generates a proton $0d_{3/2}$ spectroscopic
factor of 0.560, which is meaningfully smaller than the DOM value of 0.71 shown
to be consistent with the ($e$,$e'p$) analysis. Uncertainties arising from
choices of single-particle wave function, optical potential, and $p$-$p$
interaction do not explain this inconsistency. The inconsistency in the
spectroscopic factor suggests there is urgent need for improving the
description of $p$-$p$ scattering in a nucleus and the resulting in-medium
interaction with corresponding implications for the analysis of this reaction
in inverse kinematics.
