Vanishing of the inner index for first-order elliptic operators

Let $A$ be an elliptic differential operator of first order on a compact manifold with smooth boundary. For a collar hypersurface $\Sigma(x)$, define

$$d(x):=\dim\bigl\{u\mid Au=0\text{ and }u|_{\Sigma(x)}=0\bigr\},$$

and, for the formal adjoint $A^t$, define

$$d'(x):=\dim\bigl\{u\mid A^tu=0\text{ and }u|_{\Sigma(x)}=0\bigr\}.$$

The inner index is

$$\operatorname{ind}_0A:=d(0)-d'(0).$$

Vanishing of the inner-index conjecture. The inner index vanishes:

$$\operatorname{ind}_0A=0.$$

Vanishing of the inner index is related to weak unique continuation for $A$ and its formal adjoint and is decisive for the validity of the Bojarski conjecture. The source presents this as an open problem for general first-order elliptic operators on compact manifolds with smooth boundary.