Translated sums of primitive sets

The Erd\H{o}s primitive set conjecture states that the sum $f(A) = \sum_{a\in A}\frac{1}{a\log a}$, ranging over any $A$ of positive integers, is maximized by prime numbers. Recently Laib, Derbal, and Mechik proved translated for $f(A,h) A}\frac{1}{a(\log a+h)}$ false starting at $h=81$, comparison with semiprimes. In this note we prove such falsehood occurs already $h= 1.04\cdots$, show translate best possible We also obtain results sums $k$-almost primes larger $k$.