perverse sheaves – Decomposition of direct image of a smooth morphism, Deligne’s theorem, motives

Let $f : X to Y$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $X$ and $Y$, so use classical topology for simplicity.

I can take the constant sheaf $mathbb{Q}_X$ and (derived) push it forward to get $f_* mathbb{Q}_X in D^b_c(Y,mathbb{Q})$. There is a celebrated theorem of Deligne that $f_* mathbb{Q}_X$ is semi-simple, i.e. isomorphic to a direct sum of its cohomology sheaves. The argument uses hard Lefschetz along the fibres. (It is also true that each summand is a semi-simple local system, as a polarizable VHS, however I want to ignore that extra piece of information below.)

Suppose I replace $mathbb{Q}$ with $k := mathbb{F}_p$.

Question: Is it true that $f_* k_X in D^b_c(Y,k)$ is always semi-simple? That is, does it always split as a direct sum of its (not-necessarily semi-simple) cohomology sheaves.

I had always assumed the answer was no, but woke up this morning feeling unusually optimistic. (I have tried several times to produce a counter-example.) I understand that this is deep water, and I am happy with a heuristic answer either way (potentially using motives).