Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T_pM$, we say that $\Pi$ is non-degenerate if $$ g(X,X)g(Y,Y)-g(X,Y)^2 \neq 0.$$ Moreover, given a non-degenerate two-plane we say that it is timelike or spacelike if the above quantity is negative or positive respectively. Finally, the sectional curvature $\textrm{Sec}(\Pi)$ for a non-degenerate two plane $\Pi$ as above is defined by $$\textrm{Sec}(\Pi)=\frac{g(R(X,Y)X,Y)}{ g(X,X)g(Y,Y)-g(X,Y)^2 }.$$

**Question.** Suppose that given any non-degenerate space-like two plane in $(M,g)$ its sectional curvature is bounded from above by a fixed $K_1<0$ and that given any non-degenerate time-like two plane in $(M,g)$ its sectional curvature is bounded from below by a fixed $K_2>K_1$. Suppose that $\widetilde{g}$ is another Lorentzian metric on $M$ that is obtained by adding a sufficiently small smooth compactly supported tensor $h$ to $g$. Is it true that there exists constants $K_1'$ and $K_2'$ such that similar sectional curvature bounds hold for the perturbed metric $\widetilde g$ with the new constants $K_1'$ and $K_2'$ in place of $K_1$ and $K_2$?