What is the STP cost of preparing a Quantum Max-Cut witness?

For nonnegative rational edge weights $w_e$, let $W=\sum_e w_e>0$ and $$A=\sum_e w_e P_s^{(e)}.$$ A verifier receives one $N$-qubit witness, draws edge $e$ with probability $w_e/W$, and accepts its singlet outcome; its acceptance is $\operatorname{Tr}(A\rho)/W$. With thresholds $a>b$ separated by $1/\operatorname{poly}(N)$, distinguish $\max_\rho\operatorname{Tr}(A\rho)/W\ge a$ from $\max_\rho\operatorname{Tr}(A\rho)/W\le b$. Ask for the cost of preparing a near-optimal witness by STP, and compare the spin structure needed for preparation with what this verifier accesses.