Non-existence of non-isometric Dirac-isospectral spin lens spaces in dimensions congruent to 1 modulo 4

A spin lens space is a lens space equipped with a spin structure, and two such spaces are Dirac isospectral when their Dirac operators have the same spectrum, counted with multiplicities. Let $n$ denote their dimension.

Non-existence conjecture. Two Dirac isospectral spin lens spaces of dimension $n\equiv 1\pmod4$ are necessarily isometric.

The conjecture is motivated by the absence of examples found in the stated dimensions and the reported computational checks for bounded fundamental-group orders. It asserts that Dirac-isospectral spin lens spaces in these dimensions cannot be non-isometric.