Suppose $M$ is a compact manifold so that it admits an embedding $f:M\to N\times\mathbb{R}^l$ with a splitting of the normal bundle as $\nu_f\simeq\nu\oplus\epsilon^l$ where $\nu$ is some $k$-dimensional vector bundle and $\epsilon^l$ is the trivial $l$-dimensional bundle. The mutli-compression theorem of Rourke and Sanderson, implies that $f$ is regularly homotopy to an embedding $g:M\to N\times\mathbb{R}^l$ so that the composition

$$i:M\to N\times\mathbb{R}^l\to N$$

is an immersion with $\nu_i\simeq\nu$. For the theorem, see

Rourke and Sanderson. The compression theorem I. Geom. Topol. 5, 399-429 (2001).

or Theorem 2.2 of

Eccles and Grant. Bordism groups of immersions and classes represented by self-intersections. Algebr. Geom. Topol. 7, 1081-1097 (2007).

My question is the following. It seems to me that having $\nu_f\simeq\nu\oplus\epsilon^l$ does not necessarily imply that the trivial part if all coming from the second component of $f$ and the theorem says that we may continuously, change $f$ so that stays in the same homotopy class and all trivial parts comes from the second component of the deformed map $g$, so that collapsing all of them will not damage $\nu$ ? Does this interpretation make sense?