Cuts in models of PA and independence results

Diversity in elementary cuts

Let $M\models PA$ be countable and recursively saturated.

Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.

Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$.

Problem: How many theories of pairs $(M,K)$ are there, such that $K$ is semiregular, but not regular in $M$.

By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$.

References:

Smoryński, C. Elementary extensions of recursively saturated models of arithmetic. Notre Dame J. Formal Logic 22 (1981), no. 3, 193–203.

Smoryński, C. A note on initial segment constructions in recursively saturated models of arithmetic. Notre Dame J. Formal Logic 23 (1982), no. 4, 393–408

Kossak, Roman A note on satisfaction classes. Notre Dame J. Formal Logic 26 (1985), no. 1, 1–8.

Kaye, Richard; Wong, Tin Lok Truth in generic cuts. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005.