In Stewart's Adventures among the toroids the (T) condition is defined as follows (p. 75)
A polyhedron $\mathcal{P}$ is said to be tunnelled, or to have the property (T), if there exists a set of polyhedra $A_1,A_2,\dots,A_t$ such that $\mathcal{P}_{t+1}=\mathcal{P}$ and every $A_i$ is a tunnel or a rod.
In the nomenclature of this chapter $\mathcal{P}_{t+1}=\mathcal{P}$ means that the set $A$ must be excavated from the convex hull of $\mathcal{P}$, written $H(\mathcal{P})$, to form $\mathcal{P}$.
However this condition, as I understand it, does not seem to match how Stewart uses it. It seems to be an extraordinarily weak restriction.
It seems to rule out things that have useless excavations on the "outside" faces, for example if I excavate a pentagonal pyramid from a dodecahedron, the result is not (T). However it doesn't rule out similar useless excavations on the interior faces. For example Stewart presents a (R)(A)(Q)(T) polyhedron, which excavates a 2 square cupolae and a cube from a truncated cube:
Image by Polytope wiki user Sycamore916 CC-BY-SA-4.0
If I excavate a square pyramid from the interior faces this seems to still remain (R)(A)(Q)(T), the (R)(A)(Q) are trivial and since the polyhedron can be formed by excavating $H(\mathcal{P})/\mathcal{P}$ from $\mathcal{P}$, which increases the genus, this seems to be (T) as well. Which seems sort of wrong. I don't get why we must avoid excavations on the outside of a polyhedron but are completely fine with them on the inside.
Furthermore, it would seem that all convex polyhedra are (T), since you can just have $A=\{\}$. This certainly does not seem to be what the author means by (T). The heavy implication is that (T) implies genus $\geq 1$.
So what am I getting wrong? What is this supposed to mean?