Infinite injective dimension for rational sheaves on spaces with non-empty perfect hull

Let $X$ be a topological space with finite Cantor–Bendixson rank and non-empty perfect hull. Consider sheaves of $\mathbb{Q}$-modules on $X$. The conjecture. The injective dimension of the category of sheaves of $\mathbb{Q}$-modules over $X$ is infinite. This would establish infinite injective dimension in the stated class of spaces; the conjecture is motivated by constructing suitable sections from alternating families and disjoint subnets, but its resolution is not given here.