How efficiently can we compile into STP?

Fix Clifford+$T$ circuits with computational-basis input and output. Given a circuit $C$ of $G$ gates on $n$ logical qubits and rational $\varepsilon$, can a uniform classical compiler produce an adaptive STP protocol from maximally mixed physical qubits whose classical output distribution is within $\varepsilon$ in total variation distance of $C$? Give explicit bounds in $n$, $G$, and $\log(1/\varepsilon)$ on physical qubits, expected measurement count, a high-probability runtime bound, disjoint-pair layers, and classical processing. State the interaction graph and count every failed preparation attempt.