Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed Hodge structure, by Deligne.

Is there an example where this mixed Hodge structure does not split? In other words, where there is no isomorphism $ H^k(F(X,n), \mathbf Q) \cong \bigoplus_{i} \mathrm{Gr}^W_i H^k(F(X,n), \mathbf Q) $? I assume the answer is yes and that an example can be found already when $n=3$ and $X$ is a curve of genus $\geq 2$, but I don't know one.

I would also be interested in the analogous question for $\ell$-adic cohomology and $X$ over (say) a finitely generated field.

Motivation: I was doing some computations about cohomology of configuration spaces which led me to making an optimistic conjecture, which I then realised would imply that all mixed Hodge structures of this form must split. This made me no longer believe the conjecture, but I don't actually have a counterexample.