Uniruledness conjecture for parameter spaces of stable Higgs bundles

Let $X$ be a fixed compact Kähler manifold, let $S$ be a compact complex manifold, and let $(\mathcal{E},\Phi)$ be an effectively parametrized holomorphic family of stable Higgs bundles on $X$ parametrized by $S$.

Uniruledness conjecture. The Kodaira dimension of $S$ is $-\infty$. In particular, if $S$ is a projective manifold, then $S$ is uniruled.

This conjecture would extend the corresponding result from Riemann surfaces to higher-dimensional compact Kähler manifolds, asserting strong birational negativity for parameter spaces of effectively parametrized stable Higgs-bundle families.