Smoothing conjecture for closed cubical manifolds in codimension two

Let $N$ be a closed, oriented, cubical $n$-manifold embedded in $\mathbb{R}^{n+2}$, with $n>2$. A transverse field of 2-planes means a continuous choice of a 2-dimensional plane transverse to $N$ at every point.

Smoothing conjecture. Every such $N$ is smoothable. More precisely, $N$ admits a transverse field of 2-planes and therefore there is an arbitrarily small topological isotopy carrying $N$ onto a smooth manifold in $\mathbb{R}^{n+2}$.

This conjecture concerns smoothing higher-dimensional cubical manifolds embedded in codimension two. The stated smoothing implication uses a theorem of J. H. C. Whitehead, but the source does not provide evidence that the conjecture itself has been resolved.