Is the map $\Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$ left anodyne?

Question

This question is about objects studied in chapter 2 of Lurie's Higher Topos Theory.

Let $p : X \to S$ be a left fibration, and let $e \in X_1$ be an edge of $X$ such that $p(e)$ is an equivalence in $S$ ( - in the sense that the map $p(e) : \Delta^1 \to S$ can be extended along $\Delta^1 \to J$, where $J$ is the groupoid freely generated by $\Delta^1$). Is the edge $e$ $p$-Cartesian?

Proposition 2.4.1.5 implies that this holds true in the case that $S$ is an $\infty$-category.

ADDED

My question can be reformulated succintly to the question of whether the canonical maps $$\phi_n : \Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$$ are left anodyne, where $n \geq 2$.

Proposition. The map $\phi_2$ is left anodyne.

Proof. Push the left horn $\{0\} \to \Delta^1$ out along the map $\{0\} \to \Lambda^2_2 \sqcup_{\Delta^{\{1,2\}}} J$ that sends $0$ to $0$, and call the new endpoint $1'$, thus obtaining a 1-simplex $(0,1')$. Then fill several inner or left horns, obtaining the following simplices: $(0,1',2), (1',2,1), (1',2,1,2), (0,1',1), (0,1',1,2)$. The map $\phi_2$ is a codomain retract of the resulting map. $\ \blacksquare$

The canonical map $\phi_1 : \{1\} \to J$ is left anodyne, too, of course.

Answer 1

Today I accidentally stumbled upon Corollary 5.2.2.4 of Lurie's Higher Topos Theory:

Corollary 5.2.2.4. Let $p : X \to S$ be a Cartesian fibration of simplicial sets. An edge $e : x \to y$ of $X$ is $p$-coCartesian if and only if it is locally $p$-coCartesian.

With this corollary at hand, it is actually quite easy to answer my question:

Proposition. The maps $\phi_n : \Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$ are left anodyne, where $n \geq 2$.

Proof. Let $p : X \to S$ be a left fibration, and let $e \in X_1$ be an edge of $X$ that admits an extension $j : J \to X$ along the inclusion $\Delta^1 \to J$, (where $J$ is the groupoid freely generated by $\Delta^1$). We want to show that $e$ is $p$-Cartesian. Employing Corollary 5.2.2.4, it suffices to show that $e$ is locally $p$-Cartesian. Consider the pullback $q : X \times_S J \to J$ of $p : X \to S$ along $p \circ j : J \to S$. The map $q : X \times_S J \to J$ is a Kan fibration ( - because it is a left fibration over a Kan complex). It follows that every edge ( - and in particular the edge $e$ - ) in $X \times_S J$ is $q$-Cartesian, which implies that $e$ is locally $p$-Cartesian. $\ \blacksquare$

ADDED

Proposition. The maps $\phi_n$ are trivial cofibrations in the Joyal model structure on $\operatorname{Set}_{\Delta}$, where $n \geq 2$.

Proof. The functor $\cdot^{\flat} : \operatorname{Set}_{\Delta} \to \operatorname{Set}_{\Delta}^+$ is a left Quillen equivalence and thus preserves and reflects weak equivalences between cofibrant objects. Consider the maps $\Delta^n_n \sqcup_e J \to \Delta^n \sqcup_e J \to (\Delta^n \sqcup_e J, \{e\})$ in $\operatorname{Set}_{\Delta}^+$, where $e := \Delta^{\{n-1,n\}}$. The second map and the composition of the two maps are Joyal trivial cofibrations, so the first map is also a Joyal trivial cofibration. $\ \blacksquare$

Remark. Note that already the map $\phi_2 : \Lambda^2_2 \sqcup_{\Delta^{\{1,2\}}} J \to \Delta^2 \sqcup_{\Delta^{\{1,2\}}} J$ is not contained in the weakly saturated class of morphisms generated by the inner anodyne maps together with the maps $\{0\} \to J$ and $\{1\} \to J$: the map $p := \phi_2$ is an inner fibration that also has the right lifting property with respect to the maps $\{0\} \to J$ and $\{1\} \to J$, and $\phi_2$ does not have the left lifting property with respect to $p$.