TITLE: Extending a left fibration along an inner horn
QUESTION [11 upvotes]: Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times_{\Delta^n} \Lambda^n_i$?
The only results in that direction that I am aware of are the following two lemmas from Left fibrations and homotopy colimits by Heuts, Moerdijk:

Lemma 7.3. Consider a pullback square of simplicial sets 
  $\require{AMScd}$
  \begin{CD}
    X \times_Y Z @>g>> Z\\
    @VVV @VV p V\\
    X @>>f> Y
\end{CD}
  in which $f$ is inner anodyne and $p$ is a left fibration. Then $g$ is a trivial cofibration in the Joyal model structure.
Lemma 7.4. Let $0 < k < n$ and let $p : A \rightarrow \Lambda^n_k$ be a left fibration. Then there exists a left fibration $q : B \rightarrow \Delta^n$ and an equivalence
  \begin{CD}
    A @>g>> \Lambda^n_k \times_{\Delta^n} B\\
    @VpVV @VVV\\
    \Lambda^n_k @= \Lambda^n_k
\end{CD}
  in the covariant model structure over $\Lambda^n_k$.

REPLY [6 votes]: Yes. This is shown using minimal left fibrations in Cisinski's book Higher categories and homotopical algebra, see the proof of Theorem 5.2.10 therein. This theorem states that the simplicial set $\mathscr{S}$, whose $n$-simplices are (essentially) the left fibrations over $\Delta^n$ with small fibres, is a quasi-category.