When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?

Question

$\DeclareMathOperator{\Top}{Top} \DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of a tangent 2-field) that if $M$ has a smooth structure, the necessary and sufficient condition for it to admit two linearly independent vector fields is the following:

The Euler characteristic $\chi(M) = 0$ and the signature $\sigma(M) \equiv 0 \pmod{4}$.

Do the same two conditions characterize whether the tangent microbundle $\tau M$ of $M$ has a rank 2 trivial subbundle?

The vanishing $\chi(M) = 0$ is evidently necessary, for instance by the Lefschetz fixed point theorem; it is also sufficient for $\tau M$ to have a rank one trivial subbundle by Corollary 1.5 in Ronald J. Stern, On topological and piecewise linear vector fields.

Duane Randall has papers that work out the analogous characterization with non-orientable manifolds and manifolds of dimension $4k+1$, $4k+2$, $4k+3$, but they seem to leave the orientable 4k case unsettled in general.
Duane Randall, CAT 2-fields on nonorientable CAT manifolds
Duane Randall, On indices of tangent fields with finite singularities
Duane Randall, On 4-manifolds and span-related numbers for CAT manifolds

Let me outline the formal part of (what seems to be) the right setup. By microbundle representation theorems we can write $\tau \colon M \rightarrow B\Top(4k)$ which we want to lift along a fibration $$V_{4k,2}^{\Top} := \Top(4k) \,/\, \Top(4k,2) \rightarrow B\Top(4k,2) \rightarrow B\Top(4k) \, .$$ The primary obstruction should be the Bockstein of the $(4k-2)$nd Stiefel-Whitney class and vanish, similar to the smooth case. The next and final obstruction class will lie inside $$\co^{4k}\left(M;\pi_{4k-1}(V_{4k,2}^{\Top})\right) \cong \pi_{4k-1}(V_{4k,2}^{\Top}) \cong \pi_{4k-1}(V_{4k,2}) \cong \mathbb{Z} \oplus \mathbb{Z}/2$$ where the second isomorphism is by Theorem 2.5 of Stern's paper. The problem then becomes showing that this class always corresponds to a pair of the form $$\left(\chi(M), \,\frac{\sigma(M) \pm \chi(M)}{2} \!\!\!\mod{\!2}\right) \in \mathbb{Z} \oplus \mathbb{Z}/2 \, .$$