Below is a quote from the "Calculus of Variations and Optimal Control Theory" by D. Liberzon for context ($x^*$ denotes a local minimum):
If the set $D$ is convex, then the line segment connecting $x^*$ to an arbitrary other point $x \in D$ lies entirely in $D$. All points on this line segment take the form $x^* + \alpha d$, $\alpha \in [0, \bar{\alpha}]$ for some $d \in \mathbb{R}^n$ and $\bar{\alpha} > 0$. This means that the feasible direction approach is particularly suitable for the case of a convex $D$. But if $D$ is not convex, then the first-order and second-order necessary conditions in terms of feasible directions are conservative.
What does it mean here that the conditions in terms of the feasible directions are conservative?
My first idea would be that "conservative" is equivalent to "conditions always hold, and less demanding conditions can exist for more specific cases", but this doesn't make sense in this context. I tried to find what "conservative conditions" mean in math in general, but there seems to be no definite answer to that question.