Appendix: why can we truncate?
Let $E$ be an $n$-category, that is an $\infty$-category such that the mapping spaces $\Map(x,y)$ are $n$-truncated for every $x,y\in E$. Let $j:C\to D$ be a functor such that for every $d\in D$ the geometric realization $|C\times_D D_{d/}|$ is $n$-connected. Then for every functor $F:D\to E$ $\lim_D F$ exists if and only if $\lim_C Fj$ exists and they coincide.
Lemma 1: Let $K$ be a simplicial set such that the geometric realization $|K|$ is $n$-connected. Then for every $e\in E$ the limit of the constant functor $K\to E$ at $e$ exists and coincides with $e$.
Proof: $\mathrm{Map}(e',\lim_K e) \cong \lim_K \mathrm{Map}(e',e) = \mathrm{Map}(K,\mathrm{Map}(e',e))=\mathrm{Map}(e',e)$.
Lemma 2: Let $p:\tilde D\to D$ be a cartesian fibration such that for every $d\in D$ the geometric realization $|\tilde D_d|$ is $n$-connected. Then for every functor $F:D\to E$ the limit $\lim_D F$ exists if and only if the limit $\lim_{\tilde D} Fp$ exists and they coincide.
Proof: By an easy cofinality argument the right Kan extension along a cartesian fibration is obtained by computing the limits fiberwise. Then the thesis follows from Lemma 1.
Proof of the main result: Let $\tilde D\to D$ the cartesian fibration classified by the functor $D^{op}\to \mathrm{Cat}$ given by $d\mapsto C\times_D D_{d/}$. We have a canonical functor $C\to \tilde D$ sending $c$ to $(c,jc=jc)$. A standard cofinality argument implies that the functor $C\to \tilde D$ is coinitial. Then the thesis follows from the previous lemma.
