Prelog Chow groups of self-products of degenerations of cubic threefolds

It is unknown whether very general cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.