Does the reasoning that leads to ECH also lead to EBH?

By "ECH" I mean the Extended Cognition Hypothesis.

By "EBH" I mean something that we might call the Extended Body Hypothesis.

EBH is the hypothesis that some of the things that happen outside the boundary drawn by our skin, hair, nails and so on, are things that are happening in our body.

We could enlist something akin to the Parity Principle to argue for EBH. The Parity Principle says that: If X happens outside the head, but is such that were it done in the head, X would be cognition, then X is in fact cognition even happening as it does outside the head.

Here's a variation on the Parity Principle. If Y happens outside Z's skin/nails/hair boundary, but is such that were it done within that boundary, Y would count as Z's bodily activity, then Y is in fact Z's bodily activity even happening as it does outside the traditional body.

By "Z's bodily activity" I mean Z's normal biological functions such as heartbeats, walking, breathing, and so on.

Probably ECH is simply a special case of EBH.

My question is, if the reasoning that leads to ECH is sound, then does this suggest that a parallel line of reasoning that leads to EBH is also sound?

Would the reasoning used to arrive at ECH also suggest that Bruno's prosthetic leg in some sense really part of Bruno's body?

(A relevant text here would be Andy Clark's Natural Born Cyborgs, of course.)

I am led to ask this question because of my interest in the implications ECH has for issues in personal identity. If ECH implies (or suggests by analogous reasoning) EBH, then it seems like extending the mind must be an example of extending the self also, on both Psychologist and Somatist accounts of personal continuity. There's much more to say about that, but I'm curious to know what people think about EBH first.