Let $(M,g)$ be a globally hyperbolic Lorentzian manifold and $(E,\nabla)$ be some smooth vector bundle over $M$ equipped with a connection. Consider a generalised d'Alembertian, that is, a second order linear differential operator $$D:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$$ that up to a zero-order term coincides with the connection d'Alembertian $\mathrm{tr}_{g}(\nabla^{2})$. Now, given a spacelike Cauchy surface $\Sigma$ with normal vector field $\nu$, it is a well-known fact that the Cauchy problem $$\begin{cases}Du&=s\\u\vert_{\Sigma}&=u_{0}\\\nabla_{\nu}u\vert_{\Sigma}&=u_{1}\end{cases}$$ is well-posed for a given set of Cauchy data $(s,u_{0},u_{1})\in\Gamma^{\infty}(E)\times\Gamma(E\vert_{\Sigma})\times\Gamma(E\vert_{\Sigma})$. Furthermore, it enjoys the finite-speed-of-propagation property, i.e. $$\mathrm{supp}(u)\subset \mathcal{J}(\mathrm{supp}(s)\cup\mathrm{supp}(u_{0})\cup\mathrm{supp}(u_{1}))\tag{1}$$
Now, I stumbled over the paper (arXiv:1310.0738 [math-ph]), in which the author proves a stronger statement for a different class of hyperbolic operators, namely for symmetric hyperbolic systems, which are differential operators of first order. In this case, the author proves in Corollary 5.4 that a symmetric hyperbolic system $S:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$ enjoys the property $$\mathrm{supp}(u)\cap \mathcal{J}^{\pm}(\Sigma)\subset \mathcal{J}^{\pm}((\mathrm{supp}(Su)\cap \mathcal{J}^{\pm}(\Sigma))\cup\mathrm{supp}(u_{0}))\tag{2}$$ where $u_{0}:=u\vert_{\Sigma_{0}}$. In fact, he uses this equality to show that also $S$ has a similar finite-speed-of-propagation property as the wave operators, i.e. $$\mathrm{supp}(u)\subset \mathcal{J}(\mathrm{supp}(Su)\cup\mathrm{supp}(u_{0}))\tag{3}$$ Now, it is important to stress that (2) is a way stronger statement than (3).
Example: Take trivial intitial datum, i.e. $u_{0}=0$, and a source $s$ that is compactly supported somewhere in the future of $\Sigma$ (but not intersecting $\Sigma$), then by (2), I can conclude that $$\mathrm{supp}(u)\subset \mathcal{J}^{+}(\mathrm{supp}(s)).$$ In particular, I can conclude that $u$ has past compact support and vanishes on $\mathcal{J}^{-}(\Sigma)$. With (3), I cannot conclude that, because it just tells me that $\mathrm{supp}(u)\subset \mathcal{J}(\mathrm{supp}(s))$.
So, this leads me to the following question:
Question: Similarly to the situation for symmetric hyperbolic systems, can the finite-speed-of-propagation statement for d'Alembertians (1) be strengthened to $$\mathrm{supp}(u)\cap \mathcal{J}^{\pm}(\Sigma)\subset \mathcal{J}^{\pm}((\mathrm{supp}(s)\cap \mathcal{J}^{\pm}(\Sigma))\cup\mathrm{supp}(u_{0})\cup\mathrm{supp}(u_{1}))\quad ?$$
