  Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian
determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not
uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP
seem to satisfy properties of a Lie group. Hence, it is conjectured that a
unique diffeomorphism can be assured by its JD and curl (Uniqueness
Conjecture). In this paper, (1) an observation based on VP is derived that a
counter example to the Conjecture, if exists, should satisfy a particular
property; (2) from the observation, an experimental strategy is formulated to
numerically test whether a given diffeomorphism is a valid counter example to
the conjecture; (3) a proof of an intermediate step to the conjecture is
provided and referred to as the semi-general case, which argues that, given two
diffeomorphisms, $\pmb{\phi}$ and $\pmb{\psi}$, if they are close to the
identity map, $\pmb{id}$, then $\pmb{\phi}$ is identical $\pmb{\psi}$.
