Characterize BPI spaces that are minimal in looking down

Determine whether every Ahlfors p-regular BPI (big pieces of itself) metric space that is minimal in looking down must be p-rectifiable; equivalently, prove or refute that rectifiable BPI spaces are the only BPI spaces that are minimal in looking down in the sense of David–Semmes, where minimality means look-down equivalence with any space it looks down on.

Background

The paper shows that on rectifiable metric spaces, Lipschitz mappings admit biLipschitz pieces, which implies that rectifiable BPI spaces are minimal in looking down. Prior work established that the Heisenberg group, which is Ahlfors 4-regular and a BPI space, is not minimal in looking down, showing non-rectifiable examples can fail minimality.

This motivates a structural characterization problem: whether minimal-in-looking-down BPI spaces coincide exactly with rectifiable BPI spaces.