Final answer:
The statement is true. Immature T cells that fail to recombine the β-chain locus correctly do not pass the positive selection phase of their development and consequently die by apoptosis.
Step-by-step explanation:
The statement 'Immature T cells failing to successfully recombine a β-chain locus die by apoptosis' is true. During the development of T cells within the thymus, a process known as T cell receptor (TCR) rearrangement occurs, which includes the recombination of the β-chain locus. If an immature T cell cannot successfully recombine its TCR β-chain, it will not progress to the next stage of development, known as positive selection, where thymocytes must demonstrate the ability to bind to Major Histocompatibility Complex (MHC) proteins on thymic epithelial cells. Failure in any of these steps leads to apoptosis, a programmed cell death mechanism, which ensures that non-functional or self-reactive T cells are eliminated, preventing potential autoimmune responses.