Is the multiset of prime gaps in an admissible $ k $ -tuple of minimal diameter $ d $ determined by $ (k,d) $?

Question

Let $ (p,p+g_{1},p+g_{1}+g_{2},...,p+g_{1}+g_{2}+\cdots g_{k}) $ a prime constellation. Can the multiset $ \{g_{i}\} $ be recovered by the knowledge of the sum of its elements and its cardinal ?