The generalized arithmetic-progression divisor conjecture

Let $n$ be a positive integer. The $n$-divisor property means that every integer from $1$ through $n$ divides at least one member of a specified product. The generalized arithmetic-progression divisor conjecture. There exist $c=\widetilde{O}(\log n)$ integers $A_1,\ldots,A_c$, each of size $\widetilde{O}(n^{n^{1/3}})$, and integers $m_1,\ldots,m_c\ge0$ such that

$$1,\ldots,n\mid\prod_{i_1,\ldots,i_c=1}^{m_1,\ldots,m_c}\left(B+i_1A_1+\cdots+i_cA_c\right),$$

with $\prod_{i=1}^{c}m_i=\widetilde{O}(n^{2/3})$; equivalently, the index tuples lie in a hypercube of volume $\widetilde{O}(n^{2/3})$.

This generalizes the arithmetic-progression formulation and is proposed as a possible route to improved factorization algorithms. The source says these conjectures are unproved and gives no resolution evidence.