It is not possible to prove in general, but it is the likely "attracting" state towards which conductors evolve whenever some free charge is introduced into their inside. Evolution approaching zero free charge density can be proven only assuming specific model of the conductor, e.g. one obeying the simple Ohm law with constant uniform conductivity, and polarization $\mathbf P$ behaving as linear function of electric field (possibly zero).
Density of all charge $\rho$ can be expressed from Maxwell's equations as
$$
\rho = \epsilon_0 \nabla\cdot \mathbf E.
$$
Using the Ohm law $\mathbf j_c = \sigma \mathbf E$ where the conductivity $\sigma$ is assumed uniform, we can write this as
$$
\nabla\cdot \mathbf j_c = \frac{\sigma}{\epsilon_0} \rho.
$$
This can be easily interpreted: if there is some non-zero charge density in some small region inside the conductor, there has to be a net non-zero current out of this region. Due to this current, we expect there to be a flux of charge carriers out of the region, and magnitude of net charge density in the region to decrease in time.
This can be expressed also using the density of free charge $\rho_f$. We expect free charge (not including the bound charge that could concentrate due to non-uniform polarization and contribute to net charge density $\rho$) to be conserved, thus we have
$$
\partial_t \rho_f = - \nabla \cdot \mathbf j_c .
$$
Using also the second equation above, we can express the rate of change of free charge density $\rho_f$ as
$$
\partial_t \rho_f = - \frac{\sigma}{\epsilon_0}\rho.
$$
Thus whenever there is some region inside the metal where the net charge density $\rho$ is non-zero, free charge density changes in time in such sense as to bring net charge density towards zero (electrons will move in or out of the region, depending on sign of $\rho$).
This still allows for $\rho,\rho_f$ to behave in different ways. But assuming the metal polarization is a linear function of electric field like in dielectric media (admittedly, a big assumption), we have $\mathbf D = \epsilon \mathbf E$ with uniform $\epsilon$, and thus
$$
\rho_f = \nabla\cdot \mathbf D =\epsilon \nabla \cdot \mathbf E = \frac{\epsilon}{\epsilon_0} \rho.
$$
Thus we can write the following differential equation for the net charge density $\rho$, where $\rho_f$ does not appear:
$$
\partial_t \rho = -\frac{\sigma}{\epsilon}\rho
$$
which can be solved for any initial condition $\rho(\mathbf x,0) = \rho_0(\mathbf x)$:
$$
\rho(\mathbf x,t) = \rho_0(\mathbf x) e^{-\frac{\sigma}{\epsilon}t}.
$$
This is a remarkable result, as it shows that presence of boundaries of the metal body or their proximity to the point of interest $\mathbf x$ do not matter for the rate constant. Any non-zero net charge distribution inside the metal, provided all above assumptions hold, decays exponentially in time everywhere with the same rate constant, dependent only on the bulk properties $\sigma,\epsilon$ of the metal.
Assuming $\epsilon=\epsilon_0=8.854\mbox{e-} 12~\text{Fm}^{-1}$ and $\sigma=5.8\mbox{e-} 7~\Omega^{-1}\text{m}^{-1}$ (copper), we get relaxation time $\tau = \frac{\epsilon_0}{\sigma} = 1.5\mbox{e-}19~\text{s}$, which is 0.00015 femtoseconds. This means the transient towards zero charge density is absurdly fast, charge density dropping to $1/e$ of the initial value after only 0.00015 femtoseconds.
According to
https://web.mit.edu/6.013_book/www/chapter7/7.7.html
even in distilled water, which is not considered a good conductor, the relaxation time is still only 3.6 microseconds.
This predicted process is so fast that maybe it can break some constraints implied by special relativity, as charged particles can't move faster than light; in 0.00016 femtoseconds, anything including the charged particles can travel at most 4.5e-11 m, so at least initially, for strong enough concentration of charges, the process probably can't be exactly as fast as predicted, because the surface won't be able to take as many charged particles in arbitrarily short time. The real process thus may take longer. This means some assumption above need not be exactly satisfied in special relativistic theory, i.e. either Ohm's law or the linear relation between $\mathbf D$ and $\mathbf E$ isn't exactly obeyed, or both.
