This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.

Let $M$ be a smooth oriented manifold (without boundary) of even dimension $2k$ with $k \geq 2$. Steenrod showed that the primary obstruction for lifting the tangent bundle $\tau\colon M \rightarrow BO(2k)$ along the fibration $$V_2(\mathbb{R}^{2k}) \rightarrow BO(2k-2) \rightarrow BO(2k)$$ is $$ \beta^*(w_{2k-2}) \in H^{2k-1}(M; \pi_{2k-2}(V_2(\mathbb{R}^{2k})) = H^{2k-1}(M; \mathbb{Z})\,,$$ where $\beta^*$ is the Bockstein operator and $w_{2k-2}$ is the $(2k-2)$th Stiefel-Whitney class of $M$.

Now Theorem 2 of Massey's "On the Stiefel-Whitney classes of a manifold II" paper says that this class vanishes when $M$ is closed. Can we say the same for open $M$? If the cohomology class had field coefficients, we could argue the vanishing as in this answer and the comments under it. But here the class is integral. Can we salvage the situation using other properties the problem has (such as $\beta^*(w_{2k-2})$ being 2-torsion)?