Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits?

Let's say that a limit diagram $$\bar p:K^\triangleleft\to\def\sC{\mathscr{C}}\sC$$ is a weakly contractible limit if the simplicial set $$K$$ is weakly contractible (in that $$K\to*$$ is a weak homotopy equivalence).

I want to say for an $$\infty$$-category $$\sC$$ and an object $$x\in\sC_0$$ that the forgetful functor $$\sC_{/x}\to\sC$$ creates (i.e., preserves and reflects) weakly contractible limits. Is there a reference that already has this result?

The nLab shows here that the limit of a diagram $$p:K\to\sC_{/x}$$ coincides with the limit of the corresponding diagram $$p/x:K^\triangleright\to\sC$$, so this reduces the problem to showing that the inclusion $$K\hookrightarrow K^\triangleright$$ is an initial map (using the nLab's convention, and thus is dual to the notion of a "cofinal map" in the sense of Lurie's Higher Topos Theory). By Proposition 4.1.1.3(4) of Higher Topos Theory, this is equivalent to showing that $$K\hookrightarrow K^\triangleright$$ is left anodyne.

So perhaps an equivalent question is: if $$K$$ is a weakly contractible simplicial set, then is $$K\hookrightarrow K^\triangleright$$ left anodyne? Is there an easy way to see this?

You can also use Quillen's theorem A : to prove that $$C \to D$$ is initial, it suffices to show that for every $$d\in D$$, $$C \times_D D_{/d}$$ is weakly contractible.
In the case where $$C\to D$$ is fully faithful, this is always the case for $$d$$ in the image of $$C$$ as this pullback has a terminal object. So here we are left with proving that this is the case for the cone point, which I'm going to call $$\infty$$. For that one, as it is terminal in $$K^\triangleright$$, we see that $$K \times_{K^\triangleright} (K^\triangleright)_{/\infty} \simeq K$$ is weakly contractible, by assumption.