Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is a two-side bounded complex of finitely generated locally free $\mathcal{O}_X)$-modules.

Then we have the following definition of pseudo-coherent complex of $\mathcal{O}_X)$-modules:

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^{\bullet}$ be a complex of $\mathcal{O}_X$-modules. Let $m \in \mathbf{Z}$.

We say $\mathcal{E}^\bullet$ is $m\textit{-pseudo-coherent}$ if there exists an open covering $X = \bigcup U_i$ and for each $i$ a morphism of complexes $\alpha_i : \mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ where $\mathcal{E}_i$ is strictly perfect on $U_i$ and $H^j(\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\alpha_i)$ is surjective.

We say $\mathcal{E}^\bullet$ is $\textit{pseudo-coherent}$ if it is $m$-pseudo-coherent for all $m$.

See http://stacks.math.columbia.edu/tag/08CA

If $\mathcal{E}^\bullet$ is pseudo-coherent, then locally the cohomology $H^{\bullet}(\mathcal{E})$ is bounded above but not bounded below.

It seems that even if $\mathcal{E}^\bullet$ is pseudo-coherent, for different $m$ we may choose different cover $\{U_i\}$ and different $\mathcal{E}_i^\bullet$.

$\textbf{My question}$ is: is the definition of pseudo-coherent complex equivalent to the following:

We say $\mathcal{E}^\bullet$ is $\textit{pseudo-coherent}$ if there exists an open covering $X = \bigcup U_i$ and for each $i$ a morphism of complexes $\alpha_i : \mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ where $\mathcal{E}_i$ is a $\textit{bounded above}$ complex of finitely generated locally free sheaves on $U_i$?