For
an $\infty$-presheaf and
its $\infty$-Grothendieck construction, it ought to be the case that its projection onto the $\infty$-Grothendieck construction of its $n$-truncation is the $\infty$-Grothendieck construction on the system
of $n$-connected covers:
Is this discussed anywhere?
Here some more elaboration:
For $n \in \mathbb{N} \sqcup \{\infty\}$, we write
for the universal fibration of $n$-groupoids.
Via the naturality squares
we may canonically regard any pointed $\infty$-groupoid $(\mathcal{X}, x) \,\in\, \widehat{\mathrm{Grpd}_\infty}$ as an element in the homotopy pullback of the $n$th universal fibration along $\tau_n$.
The resulting further homotopy pullback along its classifying map is the $n$-connected cover $\mathrm{cn}_n(\mathcal{X},s)$ of $(\mathcal{X},x)$:
Since
left fibrations are preserved by pullback
a left Fibrations over the $\infty$-groupoid $\ast$ are Kan fibrations
the pullback in question is guaranteed to be an $\infty$-groupoid. Therefore, since passage to the core is a right adjoint, we may equivalently compute the $\infty$-pullback of cores.
Finally, since $Grpd_\infty$ is extensive, we may compute these pullbacks on the respective connected components of the cores.
In conclusion, this means that our $\infty$-pullback is equivalently the following:
That the top right square is a homotopy pullback follows, for instance by the pasting law and the characterization of $\infty$-actions?.
That the top left square is homotopy pullback as shown is seen by observing that it is the homotopy colimit of an evident homotopy pullback square of simplicial $\infty$-groupoids, whose right morphism is, evidently, a homotopy Kan fibration. This implies the claim by this Prop.. (The same kind of argument also applies to the bottom right square.)
Now consider the following tower of $\infty$-Grothendieck constructions on the $n$-truncations of an $\infty$-presheaf $\mathcal{X}(-)$, here shown in the first few stages:
By the pasting law, and the previous diagram, the top left square ought to be as shown, exhibiting the homotopy fibers of $P_1$ as the 1-connected covers as claimed. The same argument would apply to any $P_n$.