Anti-deSitter space vacuum, null and weak energy condition

Anti-deSitter space is characterized by a negative cosmological constant.
This implies the energy momentum tensor =
$T_{munu} = -lambda g_{munu}$, where $lambda$ is taken positive.

1. This means that the energy density $= T_{00} = rho = -lambda$ is negative. This seems to suggest that the weak energy condition is violated.
2. For a null vector $k^{mu} = (-1, 0, 0, 1)$, the quantity $T_{munu}k^{mu}k^{nu} = 0 $. This seems to suggest that the null energy condition is just satisfied.

Question: Are the above two statements correct for an Anti-deSitter vacuum? Or is my understanding wrong?
If correct, then is the null energy condition the only relevant criteria for physically valid $T_{munu}$ in Anti-deSitter space?