Let $\mathcal{C}$ be a $k$-category which we regard as an $\infty$-category whose objects all happens to be $k$-truncated. A sheaf $F$ valued in $\mathcal{C}$, in the $\infty$-categorical sense, is a presheaf satisfies the gluing condition, $$F(U) = \lim \left( \prod_{i \in I} F(U_i) \rightrightarrows \prod_{i, j \in I} F(U_i \cap U_j) \mathrel{\substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow}} \prod_{i, j, k \in I} F(U_{i} \cap U_j \cap U_k) \mathrel{\substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow}} \cdots \right) $$ where $\{U_i\}$ is an open cover of $U$ and the diagram in the limit is the standard Čech nerve of infinite length. Since $\mathcal{C}$ is a $k$-category, a reasonable guess is that one should be able to truncate at the $k$-intersections and compute a limit over a much smaller diagram while getting the same result. For example, if $\mathcal{C}$ is a $1$-category, then the gluing condition for a sheaf should be just $F(U) = \lim \left( \prod_{i \in I} F(U_i) \rightrightarrows \prod_{i, j \in I} F(U_i \cap U_j) \right)$ so we can recover the ordinary description.
My question is if this is true? If so, is there a way to write a clean proof or if there is a reference in the literature? (I couldn't find such a statement in Higher Topos theory.)
Ultimately, I care about the following case when $k = 2$ (for a Grothendieck topology): Quasicoherent sheaves $\operatorname{QCoh}$ defined in, for example, Gaitsgory and Rozenblyum's DAG, is a sheaf valued in the (very large) $\infty$-category of $k$-linear stable $\infty$-categories, for some field $k$ of characteristic $0$. For each scheme $X$, $\operatorname{QCoh}(X)$ has a standard $t$-structure and one should conclude that taking hearts $\operatorname{QCoh}^\heartsuit$ produce a sheaf of abelian categories or, in more classical terms, a stack without knowing the exact description of $\operatorname{QCoh}^\heartsuit(X)$.
This last statement should follows from my earlier question and I will be satisfied with a proof or reference for this $k=2$ case. I only asked the general question since I don't see any concept difficulty to generalize the statement to $k$. However, I wouldn't be surprised if it's hard to write an actual proof for the general case.