Determinantal point processes (abbr., DPPs) are randomly arranged points whose distribution is characterised via determinants of matrices. The entries of these matrices are given by one fixed function of two variables, K(x, y), called the kernel of the DPP. It is well-known that the kernel K is not unique and that there exist various other functions of two variables that are valid kernels of the same DPP. We refer to such kernels as equivalent kernels to K. It was recently shown by Stevens in [Random Matrices: Theory and Applications, 10(03):2150027, 2021] that, restricting to the case of symmetric kernels, all equivalent kernels of some DPP can be transformed into one another by conjugation transformations. This partially solves a conjecture of Bufetov from 2017 which states that all equivalent kernels of some DPP can be transformed into one another by conjugation and transposition transformations. In this work, we completely relax the symmetry assumptions on the kernels. We go through why this conjecture cannot hold in this general setting, but that under some surprisingly simple and natural conditions on the kernel it does.