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I'd imagine this is pretty clear cut for hobby veterans but for me the core rules FAQ's are all over the place for what is regarded as "On the battlefield".

So simple question. If I have a Phobos Captain/Lieutenant character currently in a transport that is deployed, would an infiltrator squad benefit from their reroll ability, or does the character have to have "boots on the ground" so to speak?

Generally no. Must be on the battlefield but I do think there’s exceptions but then it’s a special rule on the datasheet for the transport or the character so check them out.

And I do agree that the rules are all over the place.

Unit abilities generally are not active if you are in a Transport, unless it states otherwise. This includes things like Auras.

Anything with a aura bubble or requireing a target does nothing unless specifically stated.

Their is a grey area thats contested for rules on a datasheet that do not aura or target such as the chaplain 3+ role or admech canticals.

It depends on the wording exactly, but I doubt it. When you embark you remove the models from the battlefield.