Density conjecture for squarefree trinomial discriminant values

Let $S$ be the set of positive integers $n$ for which $n^n+(-1)^n(n-1)^{n-1}$ is squarefree. Squarefree-value density conjecture. The set $S$ has density

$$0.9934466\dots,$$

correct to that many decimal places. This conjecture predicts the density suggested by extensive computation, although the paper cannot currently prove a nontrivial lower bound for the density or even prove that infinitely many values in the sequence are squarefree.