For a complex manifold $X$ there is the Hodge filtration on cohomology, induced by the filtration on the complex of holomorphic forms given by:

$$ F^r\Omega_X^p:=\begin{cases}\{0\}\qquad\text{if }r>p\\\Omega^p_X\qquad \text{if } r\leq p. \end{cases} $$

If $X$ is compact Kähler, Hodge Theory tells us that this filtration induces a pure Hodge structure of weight $k$ on the $k$-th cohomology group. This implies, a posteriori, that morphisms respecting the hodge filtration, as given for example by the morphism $f^*$ induced on cohomology by a holomorphic map f, automatically respect it strictly, i.e. $F^r\cap im f^*=f^*F^r$.

This strictness is no longer given if we look at arbitrary maps between filtered vector spaces.

My question is if strictness still holds for morphisms induced by geometric maps if we drop the Kähler condition (and/or compactness). If not can you provide a counterexample? Is there a more general condition on the manifolds/maps that ensures strictness of induced maps?