We recently proposed a quantitative group testing (GT) scheme with low-complexity peeling decoding based on low-density parity-check (LDPC) codes. Based on finite length simulations and a density evolution analysis we were able to demonstrate that simple (dv, dc)-regular LDPC codes can be more efficient for GT than existing generalized LDPC (GLDPC) code constructions based on BCH component codes. Even larger gains were numerically observed in combination with spatial coupling. In this paper, we use vector admissible systems to prove threshold saturation and compute the corresponding potential thresholds.