The realisability conjecture for equivalence of spinor structures

Let $P$ be a principal $G$-bundle over a manifold $M$, let $u:Q\to P$ be a spinor structure, and let $\varphi:\pi_1(M)\to\pi_1(G)$ be a homomorphism. Let $u':Q'\to P$ be the spinor structure corresponding to $\varphi$.

Realisability conjecture. The bundles $Q$ and $Q'$ are equivalent if and only if $\varphi$ is realisable.

This conjecture would generalise the preceding result, which treats the special case in which all spinor structures are trivial as bundles. It proposes that the bundle type of a spinor structure is determined precisely by whether its associated homomorphism is realisable.