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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 ...

Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...