High-dimensional sequential compactness

We give examples of $n$-sequentially compact spaces that are not $(n+1)$-sequentially compact under several assumptions. We improve results of W. Kubiś and P. Szeptycki [Canad. Math. Bull. 66 (2023), 156–165] by building such examples from $\mathfrak {b=c}$ and $\diamondsuit (\mathfrak {b})+\mathfrak {d}=\omega _1$. We also introduce a new splitting-like cardinal invariant and then show that the same holds under $\mathfrak {s=b}$.