The interval-fibre conjecture for Minkowski sums of moment curves

For integers $k,n$ with $k>3$, let $\mathcal{A}_{k,n}$ denote the Minkowski sum of $n$ copies of the moment-curve segment $\{(t,t^2,\ldots,t^k)\mid -1\leq t\leq 1\}$. Let $(x_1,\dots,x_k)\in\mathcal{A}_{k,n}$. Interval-fibre conjecture. The set

$$\{y\in\mathbb{R}\mid (x_1,\dots,x_{k-1},y)\in\mathcal{A}_{k,n}\}$$

is a closed interval. The conjecture supplies the higher-dimensional analogue of the corresponding boundary property used for $k=3$; the source states that it holds for $k\leq3$, while the case $k>3$ is presented as open.