I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics:
This rather surprising result [the fact that the potential within a charge-free volume is solely dependent on the potential and its normal derivative on the boundary of the volume] is not a solution to a boundary-value problem […] since the arbitrary specification of both $\Phi$ and $\partial\Phi/\partial n$ (Cauchy boundary conditions) is an overspecification[sic] of the problem. —p.37, $\S1.8$
where the impetus for the present question was given.
The plot thickens on the next page with the exposition in $\S1.9$, given below:
It should be clear that a solution to the Poisson equation with both $\Phi$ and $\partial\Phi/\partial n$ specified arbitrarily in a closed boundary (Cauchy boundary conditions) does not exist, since there are unique solutions for Dirichlet and Neumann separately and these will in general not be consistent [emphasis mine]. This can be verified with (1.36) [see Jackson]. With arbitrary values of $\Phi$ and $\partial\Phi/\partial n$ inserted on the right hand side, it can be shown that the values of $\Phi(\mathbf x)$ and $\mathbf\nabla\Phi(\mathbf x)$ as $\mathbf x$ approaches the surface are in general inconsistent with the assumed boundary values. [boldface emphasis mine]
Here, Jackson only provides an outline of the method demonstrating that the Cauchy boundary conditions cannot be completely satisfied consistently, and does not elucidate in detail on how this might be achieved algebraically. I have thought at length about this and have, to date, not been able to come up with a satisfactory completion of the idea. Hence, I am requesting help from respondents, either by providing a full proof of the present proposition or by pointing me to resources which do so. Should you be unwilling to provide a complete answer, please at least provide a possible path forward.
TL;DR: Requesting a full proof demonstrating the fact that in general a set of Cauchy boundary conditions cannot be completely satisfied at the boundary by a scalar potential within the charge-free bounded volume without introducing inconsistencies.