  This paper focuses on investigating the density convergence of a fully
discrete finite difference method when applied to numerically solve the
stochastic Cahn--Hilliard equation driven by multiplicative space-time white
noises. The main difficulty lies in the control of the drift coefficient that
is neither globally Lipschitz nor one-sided Lipschitz. To handle this
difficulty, we propose a novel localization argument and derive the strong
convergence rate of the numerical solution to estimate the total variation
distance between the exact and numerical solutions. This along with the
existence of the density of the numerical solution finally yields the
convergence of density in $L^1(\mathbb{R})$ of the numerical solution. Our
results partially answer positively to the open problem emerged in [J. Cui and
J. Hong, J. Differential Equations (2020)] on computing the density of the
exact solution numerically.
