In "Consensus-halving via theorems of Borsuk-Ulam and Tucker", Su and Simmons describe a way to divide a cake to two parts such that each of $n-1$ people believe the parts have the same value. This can be done with $n-1$ cuts ($n$ pieces), which is optimal.
Once we have such a division, we can let the $n$-th person choose which half is better. Now, regardless of how the groups are divided, all people in each group believe that the value of their group is at least $1/2$. So we have a group division in which the number of pieces is equal to the total number of people, which is optimal.
So for 2 groups, the problem is solved. The case of $k>2$ groups is mentioned as a conjecture in their Open Problems section.