  We prove that, at the mini superspace level, and for an arbitrary Brans-Dicke
parameter, one cannot tell traditional Einstein-Hilbert gravity from local
scale invariant Weyl-Dirac gravity. Both quantum mechanical cosmologies are
governed by the one and the same time-independent single-variable
Hartle-Hawking wave function. It is only that its original argument, the cosmic
scale factor $a$, is replaced by $a\phi$ ($\phi$ being the dilaton field) to
form a Dirac in-scalar. The Weyl vector enters quantum cosmology only in the
presence of an extra dimension, where its fifth component, serving as a 4-dim
Kaluza-Klein in-scalar, governs the near Big Bang behavior of the wave
function. The case of a constant Kaluza-Klein in-radius is discussed in some
detail.
