Fix a category or $\infty$-category $C$ with all small limits.
We call a natural transformation $\alpha\colon f\to g$ between two functors $f,g\colon K\to C$ Cartesian, if for every arrow $k\colon a\to b$ in $K$, the naturality square of $\alpha$ over $k$ identifies $f(a)$ with the fiber product $g(a)\times_{g(b)}f(b)$, i.e. every naturality square of $\alpha$ is a pullback square in $C$.
Denote by $\overline K$ the cone on $K$ (i.e. $K$ with an additional initial object $\star$). For every diagram (a.k.a. functor) $f\colon K\to C$ we denote by $\overline f\colon\overline K\to C$ the right Kan extensions of $f$ along the inclusion $K\hookrightarrow \overline K$; in other words, $\overline f$ is a cone that exhibits $f(\star)\in C$ as the limit of the diagram $f$.
The limit cone $\overline f\colon\overline K\to C$ is called absolute, if for every functor $F\colon C\to D$, the composition $F\circ \overline f\colon \overline K \to D$ is still a limit cone.
Given a natural transformation $\alpha\colon f\to g$ between two diagrams $f,g\colon K\to C$, there is a unique transformation $\overline \alpha\colon \overline f\to\overline g$ which restricts to $\alpha$ and is induced by the universal property of the limit cone.
Question: Let $\overline f,\overline g\colon \overline K\to C$ be absolute limit cones extending diagrams $f,g\colon K\to C$. If $\alpha\colon f\to g$ is a Cartesian transformation, does it follow that $\overline \alpha\colon\overline f\to \overline g$ is also Cartesian?
While I am interested in the general statement, I am mostly concerned about the special case where $K$ is the simplex category $\Delta$, hence $\overline K$ is the augmented simplex category $\Delta_+$.
