Subhomomorphism relation between transfer condition operator and intermediately operator over composition operator

Statement
Let $$T$$ and $$I$$ denote the fact about::transfer condition operator and the fact about::intermediately operator on subgroup properties, and let $$*$$ denote the composition operator. Then, for subgroup properties $$p$$ and $$q$$, we have:

$$I(p) * T(q) \le I(p * q)$$.