Parity-vanishing surjectivity conjecture for Hilbert-manifold cobordism

Let $X$ be a separable Hilbert manifold, and let

$$\rho:\mathcal U^*(X)\longrightarrow\displaystyle\lim_{\overleftarrow{M\downarrow X}}MU^*(M)$$

be the homomorphism obtained by restricting classes along all proper smooth maps from finite-dimensional manifolds $M$ to $X$. Write $\mathcal U^{\operatorname{ev}}(X)$ and $\mathcal U^{\mathrm{odd}}(X)$ for the even and odd parts of $\mathcal U^*(X)$.

Parity-vanishing surjectivity conjecture. If $\mathcal U^{\operatorname{ev}}(X)=0$ or $\mathcal U^{\mathrm{odd}}(X)=0$, then $\rho$ is a surjection.

This is a conditional refinement of the general inverse-limit surjectivity claim. The source gives it as a reasonable conjecture consistent with the examples under consideration; no resolution is stated.