  Fisher's information measure plays a very important role in diverse areas of
theoretical physics. The associated measures as functionals of quantum
probability distributions defined in, respectively, coordinate and momentum
spaces, are the protagonists of our present considerations. The product of them
has been conjectured to exhibit a non trivial lower bound in [Phys. Rev. A
(2000) 62 012107]. We show here that such is not the case. This is illustrated,
in particular, for pure states that are solutions to the free-particle
Schr\"odinger equation. In fact, we construct a family of counterexamples to
the conjecture, corresponding to time-dependent solutions of the free-particle
Schr\"odinger equation. We also give a new conjecture regarding any
normalizable time-dependent solution of this equation.
