This paper introduces a novel framework for open-set speaker identification in household environments, playing a crucial role in facilitating seamless human-computer interactions. Addressing the limitations of current speaker models and classification approaches, our work integrates an pretrained WavLM frontend with a few-shot rapid tuning neural network (NN) backend for enrollment, employing task-optimized Speaker Reciprocal Points Learning (SRPL) to enhance discrimination across multiple target speakers. Furthermore, we propose an enhanced version of SRPL (SRPL+), which incorporates negative sample learning with both speech-synthesized and real negative samples to significantly improve open-set SID accuracy. Our approach is thoroughly evaluated across various multi-language text-dependent speaker recognition datasets, demonstrating its effectiveness in achieving high usability for complex household multi-speaker recognition scenarios. The proposed system enhanced open-set performance by up to 27\% over the directly use of efficient WavLM base+ model.