Unitary inequivalence of representations from orthogonal vacua

Let $|0\rangle_p$ and $|0\rangle_q$ be the vacuum states in the Hilbert spaces $L^2_p({\cal F})$ and $L^2_q({\cal F})$, respectively, associated with the Sobolev parameters $p$ and $q$. Let the holonomy-diffeomorphism algebra act in the corresponding representations. Orthogonal-vacua conjecture. If

$${}_q\langle 0\mid 0\rangle_p=0,$$

then the associated representations of the holonomy-diffeomorphism algebra are unitarily inequivalent. The preceding discussion establishes orthogonality for distinct parameters in the stated construction, but no resolution of the representation-theoretic implication is given.