Transfer condition operator preserves intersection-closedness

Statement

Suppose $T$ is the Transfer condition operator (?). Then:

• If $p$ is a Finite-intersection-closed subgroup property (?), so is $T(p)$.
• If $p$ is an Intersection-closed subgroup property (?), so is $T(p)$.

Related facts

• Subhomomorphism relation between transfer condition operator and intersection operator
• Intersection-closure operator preserves transfer condition