Local square-integrability conjecture for first-order differential operators

Let $M$ be a Riemannian manifold, let $E$ be the relevant vector bundle, and let $T:C_0^\infty(E)\to C_0^\infty(E)$ be a first-order differential operator. Assume that $T^2$ is essentially self-adjoint. For $u\in L^2_{\operatorname{loc}}(E)$ satisfying

$$T^2u\in L^2_{\operatorname{loc}}(E),$$

the expression $Tu$ is locally square-integrable.

Local regularity conjecture. Under these assumptions,

$$Tu\in L^2_{\operatorname{loc}}(E).$$

This implication would establish the local integrability condition needed in the essential self-adjointness result for Schrödinger operators of the form $D^tD+V$. The source presents it as a conjecture and gives no resolution, so its status remains open.