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Are there any known examples of index 1 smooth Fano 4-folds X with $\mathsf{Pic}(X)\cong\mathbb{Z}$ with $h^2(\Omega^1_X)\neq0$?

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Did you already try complete intersections of theta divisors in a moduli space of principal $G$-bundles ($G semisimple) on a smooth, projective curve? The weight 1 Hodge structure of the curve induces a weight 3 Hodge structure of the moduli space, which maps to the weight 3 Hodge structure of the complete intersection. –  Jason Starr May 17 '11 at 20:54
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