For MIMO systems with binary or QPSK signalling, soft demodulation techniques based on semidefinite relaxation (SDR) offer a desirable balance between performance and computational cost. In this paper we expand on an extension of that approach to systems with higher-order QAM signalling. In particular, we develop a customized dual-scaling algorithm to solve the semidefinite program that constitutes the core computational task of the demodulator, and we evaluate its performance via simulations and using an EXIT chart analysis. In comparison to several state-of-the-art demodulators, the proposed demodulator provides performance that comes close to that of the list sequential (LISS) demodulator, at a computational cost that is close to that of the minimum mean square error (parallel) soft interference cancellation (MMSE-SIC). Furthermore, the distribution of the computational cost is concentrated around its mean.