Worked solution: Kang–Kelly–Kühn–Methuku–Osthus asymptotic proof via iterative absorption (2021)

Kang, Kelly, Kühn, Methuku and Osthus (2023, Section 1.2, Theorems 1.2 and 1.3) prove a quantitative stability result confirming Kahn's prediction. Theorem 1.2 states that for every $\delta > 0$, there exist $n_0, \sigma > 0$ such that for $n \ge n_0$, any $n$-vertex linear hypergraph $\mathcal{H}$ with maximum degree $\Delta(\mathcal{H}) \le (1-\delta)n$ and at most $(1-3\delta)n$ edges of size $(1\pm\delta)\sqrt{n}$ satisfies $\chi'(\mathcal{H}) \le (1-\sigma)n$.

In fact, stability is not merely a bonus corollary: inside the main proof, whenever $\mathcal{H}$ is far from the finite projective plane (handled in Section 5) and has no vertices of degree close to $n$ (handled by absorption and $1$-factorisation in Sections 4 and 7), the slack $(1-\sigma)n$ is precisely what gives the nibble and locally sparse colouring steps enough breathing room to finish without running out of colours.