Even-dimensional stacky Euler-class formula with Catalan correction factor

Let $n\geq 2$ be even, let $\pi^*V$ be the pullback of the vector bundle to the stack considered in the paper, and let $\mathbb{H}=\langle 1\rangle+\langle -1\rangle$ denote a hyperbolic plane. Let $c(n-1)$ be the $(n-1)$st Catalan number and let $i(n)$ be the correction factor from the preceding conjecture. Stacky Euler-class conjecture. For every even $n\geq 2$,

$$e(\pi^*V)=\frac{c(n-1)-i(n)}{2}\mathbb{H}+i(n)\langle 1\rangle.$$

This extends the explicit computation $e(\pi^*V)=2\mathbb{H}+\langle 1\rangle$ at $n=4$ and predicts a uniform formula for the Euler class on the root-stack construction; the correction factor remains unspecified.