  We study the nonnegativity of stringy Hodge numbers of a projective variety
with Gorenstein canonical singularities, which was conjectured by Batyrev. We
prove that the $(p,1)$-stringy Hodge numbers are nonnegative, and for
threefolds we obtain new results about the stringy Hodge diamond, which hold
even when the stringy $E$-function is not a polynomial. We also use the
Decomposition Theorem and mixed Hodge theory to prove Batyrev's conjecture for
a class of fourfolds.
