The Lazy Caterer's sequence conjecture for restricted fractional partitions

Let $j>2$ and let $k,j\in\mathbb{N}$. For a sequence of numerators $\{n_j\}$, define

$$G=\frac{n_1}{j}+\frac{n_2}{j}+\cdots+\frac{n_{j-1}}{j}+\frac{n_j}{j}.$$

A sequence enables solutions when every positive integer $k<\lfloor G\rfloor$ can be represented as a sum of distinct fractions with denominator $j$ and numerators from the sequence. The Lazy Caterer's sequence conjecture. The Lazy Caterer's sequence (A000124) enables solutions for all $k<\lfloor G\rfloor$, with $j>2$ and $k,j\in\mathbb{N}$. This also appears to be true for the Cake Numbers (A000125).