R. Guy, Unsolved problems in number theory, 3rd edition, Springer, 2004.

In this book, on page 167-168, Problem C5, Sums determining members of a set, discusses a question Leo Moser asked: suppose $X\subset\mathbb{Z}$ is an $n$-element set and $A$ be set of all $k$-element sums of subsets of $X$. Is there any other $Y\neq X$ such that $Y\subset\mathbb{Z}$ such that its $k$-element sums are exactly $A$?

Only small cases were known: $k=2,3,4,5$ and any $n$.

Question. What is the general status of this problem for $k>5$?

Update. Max Alekseyev provided some references. I am still interested in the question: are there (non-cosmetic) equivalent formulations of the above problem?