Erdős Problem #707 — Embedding Finite Sidon Sets in Perfect Difference Sets

Let $A\subseteq\mathbb{N}$ be finite and Sidon, where Sidon means that for all $a_1,a_2,b_1,b_2\in A$, if $a_1+a_2=b_1+b_2$, then either $a_1=b_1$ and $a_2=b_2$, or $a_1=b_2$ and $a_2=b_1$. Does there exist a set $B\subseteq\mathbb{N}$ and a natural number $n>0$ such that $A\subseteq B$ and the ordered differences $b_1-b_2$ of distinct elements $b_1,b_2\in B$ represent every nonzero residue modulo $n$ exactly once; that is, is $B$ a perfect difference set modulo $n$?