Can every element of a homotopy group of a smooth manifold be represented by an immersion?

Let $$M$$ be a smooth manifold of dimension $$n$$. Let $$\alpha \in \pi_i (M)$$, for some $$i \leq n$$, and $$f:S^i \to M$$ be a map of the sphere representing the homotopy class $$\alpha$$. Can $$f$$ always be chosen to be an immersion? I.e. does every homotopy class of the maps from the sphere $$S^i$$ to $$M$$ contain an immersion?

This feels like it should be a classical result but I don't know much about this, nor do I have any idea how to approach such a problem. If it is true, could I get a reference to the result, or if it's simple, a hint for the solution? Or is there some subtlety that I'm missing that makes this statement false?

• Certainly not in general when $i=n$. Then an immersion is covering map. I don't know what happens for, say $i=n-1$. Nov 6, 2020 at 2:11
• On the other side, for sufficiently large codimension ($i\le\lfloor n/2\rfloor$ I believe) such an immersion exists by approximation/transversality arguments. Nov 6, 2020 at 2:27
• Asked and answered on MO: mathoverflow.net/questions/375919/… Jan 7, 2021 at 10:31