The Pell-equation existence conjecture for $(1,1)$-antimagic graph unions

Let $P_3$ be the path on three vertices. For a graph $H$, a $(1,1)$-antimagic labeling is an antimagic labeling whose vertex sums are exactly $1,2,\ldots,|V(H)|$. Pell-equation existence conjecture. Let $m$ and $n$ be positive integers satisfying

$$(2n+1)^2-2(2m+1)^2=-1.$$

Then there exist a graph $G$ and an integer $c$ such that

$$|V(G+cP_3)|=n,\qquad |E(G+cP_3)|=m,$$

and $G+cP_3$ has a $(1,1)$-antimagic labeling. The preceding discussion motivates this as an existence question for graphs with prescribed numbers of vertices and edges satisfying the Pell equation, especially because components isomorphic to $P_3$ are forced in the relevant setting; the supplied text does not state that the conjecture has been resolved.