I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though sadly I have not enough background to understand the answer. Anyway, fixed a scheme $X$ and following the definitions of Thomason-Trobaugh, I call a complex $E^{\cdot}$ of $\mathcal{O}_X$-modules strict-$n$-pseudocoherent if it is bounded above and $E^k$ is locally free of finite rank in each degree $k\geq n$. I call it strict pseudocoherent if it's strict-$n$-pseudocoherent for any integer $n$. I call it $n$-pseudocoherent if for every $x\in X$ there exists a neighborhood $U$, a strict $n$-pseudocoherent complex $F^{\cdot}$ of $\mathcal{O}_U$-modules and a quasi-isomorphism $F^{\cdot}\to E^{\cdot}|_U$ (birefly, $E^{\cdot}$ is locally quasi-isomorphic to a strict $n$-pseudocoherent complex). Eventually, $E^{\cdot}$ is said to be pseudocoherent if it is $n$-pseudocoherent for every integer $n$.
Now, clearly for a pseudocoherent complex and a fixed $x$, there is for any $n$ an open neighborhood of $x$, depending on $n$, over which it's quasi-isomorphic to a strict $n$-pseudocoherent complex. It seems reasonable that it could not exists an open neighborhood of $x$ such that restricted to it the complex is locally quasi-isomorphic to a strict $n$-pseudocoherent for every $n$ (basically, because the intersection of the previous ones could no longer be open). But how to provide a concrete example of this happening?
Now, as far as I know (from the paper of Thomason-Trobaugh, Proposition 2.3.1) we cannot expect to find an example among complexes of quasi-coherent $\mathcal{O}_X$-modules. My attempt (which I realize to be quite wrong) is here: https://math.stackexchange.com/questions/3950352/example-of-pseudocoherent-complex-which-is-not-locally-quasi-isomorphic-to-a-str, where I was looking for an example over the affine line, considering a complex of skyscraper sheaves. Don't really know whether it can work with suitable adjustments or whether the answer should be found in a different context.
Any help is appreciated, thanks in advance!
