The sharp $L^1$ inequality for sup-convolution

Let $C\subset\mathbb{R}^k$ be a compact convex domain and let $f:C\to\mathbb{R}$ be bounded and measurable. Write $f^{\ast n}$ for the $n$-fold sup-convolution of $f$, let $\operatorname{co}(f)$ denote its convex envelope, and let $A(k,\ell)$ be the Eulerian number counting permutations of $S_k$ with $\ell$ descents. Sharp $L^1$ inequality for sup-convolution. For arbitrary $k,n\geq 1$,

$$\int_C\bigl(f^{\ast n}(x)-f(x)\bigr)\,dx\geq c_{k,n}\int_C\bigl(\operatorname{co}(f)(x)-f(x)\bigr)\,dx,$$

where

$$c_{k,n}=\frac{1}{n^k}\sum_{m=1}^k\frac{n-m}{n}\binom{n+k-m}{k}A(k,m-1)=\frac{k+1}{n^{k+1}}\bigl(1^k+\ldots+(n-1)^k\bigr).$$

This conjecture extends the sharp inequality proved in the paper for the cases treated there; the general validity for arbitrary dimensions and convolution powers remains open.