Lehn's conjecture on Segre-number generating series for tautological bundles

Let $S$ be a smooth projective surface and let $L$ be a line bundle on it. Define

$$N_n=\int_{S^{[n]}}s_{2n}(L^{[n]}).$$

Here $H$ is the divisor corresponding to $L$, $K$ is the canonical divisor, and

$$a=HK-2K^2,\qquad b=(H-K)^2+3\chi(\mathcal{O}_S),\qquad c=\frac{1}{2}H(H-K)+\chi(\mathcal{O}_S).$$

Let $k\in\mathbb{Q}[[z]]$ be the formal power series inverse of

$$z=\frac{k(1-k)(1-2k)^4}{(1-6k+6k^2)^3},$$

so that $k=z-9z^2+94z^3-\cdots$. Lehn's conjecture. The generating series of the numbers $N_n$ is

$$\sum_{n\geq 0}N_nz^n=\frac{(1-k)^a(1-2k)^b}{(1-6k+6k^2)^c}.$$

This conjecture gives an explicit universal expression for the generating series of top Segre integrals of tautological line bundles on Hilbert schemes of points on surfaces. The source notes that the conjecture was still open at the time of writing, although the rank $1$ and rank $-1$ cases of the paper's related conjecture are established.