A relative index theorem too good to be true?

A few days ago I was reading with Rudolf Zeidler the recent preprint by Zhizhang Xie (arXiv:2103.14498). We were especially focusing on the proof of the relative index theorem therein and the global analytic lemmas leading to it. While doing this, we realized that we can not come up on the spot with a counter-example to the following extremely strong relative index problem (which is a priori much stronger than the theorem claimed by Zhizhang Xie):

Let \(Z\) be a closed Riemannian manifold and let \(D_1,D_2\) be two operators of Dirac type acting on the same (graded) hermitian vector bundle over \(Z\). Assume that \(D_1\) and \(D_2\) coincide on an open subset of \(Z\). Must then their Fredholm indices coincide?

By an operator of Dirac type we mean a (graded) first-order, symmetric differential operator \(D\) whose symbol satisfies \(\sigma_D(x,\xi)^2 u = -\|\xi\|^2 u\). Of course, one can also ask the same question more generally for (graded) first-order, symmetric and elliptic differential operators. And one can also make the conclusion much stronger by asking whether even their higher index classes in \(KO_*(C^*_{\max}(\pi_1 Z))\) must coincide.

The tricky part in the above problem is that we assume that \(D_1\) and \(D_2\) act on the very same (graded) hermitian vector bundle. If we instead just demand that we have a bundle isometry over the open subset intertwining the operators, then we know counter-examples: for example, one can use those pairs of bundles and operators acting on them which are used to prove that compactly enlargeable manifolds do not admit Riemannian metrics of positive scalar curvature.

Do you know a counter-examples to the above problem (or want to say anything else about it)? Then just write us an eMail or post a comment further below.