Erdős's extension conjecture for finite Sidon sets

A finite Sidon set is a finite set of integers whose differences of distinct elements are all distinct. A finite perfect difference set is a finite set whose differences represent every nonzero residue modulo an appropriate modulus exactly once. Erdős's extension conjecture. Every finite Sidon set can be extended to a finite perfect difference set.

The conjecture would imply Erdős's density conjecture for Sidon sets. The source reports that the version requiring the modulus to have the form $p^2+p+1$ with $p$ prime is false, via the counterexample $\{1,2,4,8\}$; the unrestricted formulation above is the conjectural statement considered in the paper.