The structure of the double discriminant

For a polynomial f(x) = ∑ᵢ₌₀ⁿ aᵢ xⁱ, we study the double discriminant DDn,k = discaₖ discₓ f(x), which appears in the proof of the van der Waerden--Bhargava theorem. We conjecture that DDn,k is the product of a square, a cube, and possibly a linear monomial and we prove this when $k=0$. We also investigate the (typically large and smooth) outlying integer constant in the factorization of DDn,k.