The generalized projected hypercube conjecture for difference sets

Let $b$ be a positive integer, and let $g(n,k,\ell)$ denote the relevant extremal function for difference sets. For the recursively constructed $b$-ary projected-hypercube sets, the construction has cardinality $b^i$ and difference-set size $((2b-1)^i-1)/2$. Generalized projected hypercube conjecture. When $k$ is a power of $b$,

$$g\left(n, k, \frac{k^{\log_b(2b-1)} +1}{2} \right) = O\left(k^{\log_b(2b-1)} \right).$$

This is proposed as the natural generalization of the corresponding result of FLS19; the paper presents it as a conjecture in the future-work discussion, and no proof or resolution is supplied there.