Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold.

In Atiyah-Singer "the index of elliptic operators I" pg 518, it is stated that:

theindex of $\mathcal D$ as a Fredholm operator, depends just on the homotopy class of the symbol of $\mathcal D$, $\sigma(\mathcal D):SX\to Iso(E,F)$ in the space of continuous invertible symbols of order $k$.

I have no problem in showing that if two operators have homotopic symbols then they are also homotopic as Fredholm operators and therefore they have the same index. The converse seems to be harder to prove and I'm starting to suspect that it is false.

Q. Suppose that two elliptic operators $\mathcal{D_1},\mathcal {D_2}:\Gamma(E)\to \Gamma(F)$ of the same order have the same index. Are $\sigma(\mathcal{D_1}) $ and $\sigma(\mathcal{D_1})$ homotopic?

For example, if I am not wrong, when $\dim X$ is odd, then all elliptic operators have index zero. However if we consider $E=F$ to be a trivial bundle, then homotopy classes of symbols should correspond to homotopy classes of maps $[SX, GL(\mathbb{C},r)]$ and this set is not a singleton in general.

P.S. I have posted this on math.stackexchange.com but I did not get any answer or comment.