Boyd et al.'s density conjecture for square-free trinomial discriminants

Let

$$S=\left\{n\in\mathbb N:n^n+(-1)^n(n-1)^{n-1}\text{ is square-free}\right\}.$$

Here a positive integer is square-free if it is not divisible by the square of any prime, and the natural density of $S$ is the limiting proportion of positive integers in $S$.

Boyd et al.'s conjecture. The set $S$ has density

$$0.9934466\ldots,$$

with the displayed decimal correct to that many decimal places.

The quantity is, up to sign conventions, the discriminant of the trinomial $x^n-x-1$. The source reports computational evidence and sporadic nonsquare-free values, but does not give a proof of the asserted density.