In this paper, we address the problem of data reconstruction from privacy-protected templates, based on recent concept of sparse ternary coding with ambiguization (STCA). The STCA is a generalization of randomization techniques which includes random projections, lossy quantization, and addition of ambiguization noise to satisfy the privacy-utility trade-off requirements. The theoretical privacy-preserving properties of STCA have been validated on synthetic data. However, the applicability of STCA to real data and potential threats linked to reconstruction based on recent deep reconstruction algorithms are still open problems. Our results demonstrate that STCA still achieves the claimed theoretical performance when facing deep reconstruction attacks for the synthetic i.i.d. data, while for real images special measures are required to guarantee proper protection of the templates.