Logarithmic Gysin sequences for regular immersions

For a regular immersion of schemes $Z\to X$ and a cohomology theory of fs log schemes, we formulate the logarithmic Gysin sequence using the “logarithmic compactification” $({\mathrm{B}\mathrm{l}}_{Z}X,E)$ instead of the open complement $X-Z$, where $E$ is the exceptional divisor. We show that all ${\mathbb{A}}^1$-invariant cohomology theories produced from motivic spectra and various non ${\mathbb{A}}^1$-invariant cohomology theories like Nygaard completed prismatic cohomology admit logarithmic Gysin sequences.