Conjecture on products of 2-wise uniform distributions

Let $H$ be a group, let $t$ be a positive integer, and let $X,Y$ be independent random variables that are 2-wise uniform over $H^{t}$, meaning that every pair of coordinates of each variable is uniformly distributed over $H^{2}$. Product conjecture. The product $XY$ cannot have a fixed value; equivalently, $XY$ is not supported on a single element of $H^{t}$. The question is posed in the context of understanding how products of partially uniform distributions increase independence, and the source does not provide a resolution.