In the appendix of the book of R. Kiehl and R. Weissauer, Weil Conjectures, Perverses Sheaves and l'adic Fourier transform, they quoted the following theorem.
Let $X$ be a finitely generated scheme over a separably closed $k$, and $\mathcal{F}$ an étale sheaf on $X$, then $H^v(X,\mathcal{F})=0$ for $v > 2\dim(X)$. This bound can be sharpened to $v > \dim(X)$ if $X$ is affine.
My confusion is that do we require $\mathcal{F}$ to be a constructible (or torsion after passing to limit) sheaf or just an arbitrary sheaf? When I check other books like James Milne, Etale Cohomology or Leifu, Etale Cohomology Theory, I often see that they require constructibility for this result. Let me take Fu's book for instance.
Theorem 7.5.5 (Lei Fu). Let $X$ be a finitely generated scheme over a separably closed field $k$, $n$ an integer and $\mathcal{F}$ a torsion sheaf on $X$ such that $\mathcal{F}_{\overline{a}}=0$ for any point $a \in X$ with $\dim \overline{\left \{a \right \}} > n$. Then $H^v(X,\mathcal{F})=0$ for any $v > 2n$. In particular, $H^v(X,\mathcal{F}) = 0$ for any $v > 2\dim(X)$.
The first step reduces $\mathcal{F}$ to a constructible sheaf by a routine argument and then proceed by induction on $n$ (which maybe independent of $\dim(X)$ but of course less than $\dim(X)$). But I can see the only point where constructibility is used is the first part of Theorem 7.5.5. More concretely, the first induction $n=0$ requires the following condition $$\dim \mathrm{supp}\mathcal{F} \leq n$$ which seems only true when $\mathcal{F}$ is constructible (see the proof of 7.5.1 in the same book). But for the particular case $H^v(X,\mathcal{F})=0$ for $v>2\dim(X)$ ($n = \dim(X)$ in this case), this condition always hold true so I think we can ignore constructibility in this particular statement. Is this right?