By [GJKM19, Proposition 1.2] we have $\varphi=\varphi\circ p_{\mathfrak{X}}$. As in the previous result $\varphi$ is convex on $\tau$ and

$$\mu=c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})=\Deg(S)\cdot\MA(\varphi)$$

on $\tau$. But a solution to the archimedean Monge-Ampère problem is given by a second antiderivative of $f$ and the solution is unique up to addition of a linear function. Hence $\varphi\in C^{k+2}(\tau)$ and $\Deg(S)\cdot\varphi^{\prime\prime}=f$. ∎