Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction

We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. We prove an invariance principle (functional central limit theorem) under almost every fixed environment. The assumptions are nonnestling, at least two spatial dimensions, and a $2+\epsilon$ moment for the step of the walk uniformly in the environment. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.