Difference-set graceful-labeling conjecture for regular connected graphs

Let $D$ be a $(v,k,\mu)$ difference set in a group $G$, and define

$$\lambda_i=\frac{\mu i}{\gcd(k-1,\mu)}$$

for $1\leq i\leq\gcd(k-1,\mu)$. A graph is $D$-graceful when its vertices can be labeled by the elements of $D$ so that the differences associated with its edges are evenly distributed as required for a difference graph. Difference-set graceful-labeling conjecture. Every connected regular graph $\Gamma$ of order $k$ and degree

$$\frac{(k-1)i}{\gcd(k-1,\mu)}$$

is $D$-graceful; consequently, there exists a $(v,\Gamma,\lambda_i)$ difference graph with vertex set $D$. The conjecture is motivated by the absence of counterexamples for regular connected graphs of order $|D|$; the source records no general proof, although it notes later special cases for Paley difference sets and circulant graphs.