Logarithmic Gysin sequences for regular immersions

For a regular immersion of schemes Z→ X and a cohomology theory of fs log schemes, we formulate the logarithmic Gysin sequence using the "logarithmic compactification" (BlZ X,E) instead of the open complement $X-Z$, where E is the exceptional divisor. We show that all A¹-invariant cohomology theories produced from motivic spectra and various non A¹-invariant cohomology theories like Nygaard completed prismatic cohomology admit logarithmic Gysin sequences.