Pyber's logarithmic lower-bound conjecture for subgroup coverings of finite abelian groups

Let $A$ be a finite abelian group, and let $g(A)$ denote the minimum number of subgroups in an irredundant subgroup covering of $A$ whose intersection is trivial, with $g(A)=\infty$ if no such covering exists.

Pyber's conjecture. There exists a fixed constant $c>1$ such that

$$g(A)>\log_c|A|$$

for every finite abelian group $A$.

This asks for a logarithmic lower bound on the number of subgroups needed to cover finite abelian groups irredundantly with trivial intersection. The source gives no resolution, so the conjecture is treated as open.