Such a restriction requires a change in the foundations of quantum mechanics or its interpretation.
The Heisenberg's uncertainty principle is an inequality:
$\sigma_p\cdot \sigma_x \geqslant \hbar/2$; as such, it doesn't forbid a system to be in the state where the uncertainty of the momentum and coordinate are directly related by, say, being inversely proportional to each other in a way that makes it an equality:
$$\sigma_x = \frac{\hbar}{2\sigma_p}.$$
However, if one keeps only states which satisfy this relation, one arrives at a contradiction with other principles of quantum mechanics. The condition above severely restricts the variety of quantum mechanical states a system can be in; for instance, the harmonic oscillator could only exist in what is called coherent states and cannot exist in states with definite energy; for a hydrogen atom, no stationary state (with a fixed energy level) is possible either. But the measurement process of, say, energy should put the system it into a state with definite energy; however these states are now forbidden. Thus, to stay consistent, one has to review at least the part of the foundations of quantum mechanics which tells what a measurement does to the particle.
Any other direct functional connection between uncertainties should lead to similar problems, because it leads to too few allowed states to be compatible with the existing interpretations of measurement in quantum mechanics.