I suspect it isn't true as stated unfortunately.  If you really need an example, I can try to cook one up.

Probably it is true for something like $P^0$, for a definition see [Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems][1]  (there are some modifications one can make to that definition too which might make this easier).

Definitely it is true for $P^0$ under suitable positivity assumptions.  What can you assume about $M - K_X - \Delta$?.  


  [1]: http://arxiv.org/abs/1212.6956