Regularity of successor thorns under large-cardinal assumptions

Let \textit{\th}_0=\omega, let \textit{\th}_{\alpha+1}=\textup{\TH}(\textit{\th}_\alpha^\omega), and for limit $\gamma$ let \textit{\th}_\gamma=\bigcup_{\alpha<\gamma}\textit{\th}_\alpha. Assume sufficient large cardinals and $\mathrm{V}\ne\mathrm{L}(X^\omega)$ for every set $X$. Regularity conjecture. For every ordinal $\alpha>0$,

\textit{\th}_{\alpha+1}=\textit{\th}_\alpha^+

is regular. This anticipated calculation of the thorn sequence is used in the paper's programme of obtaining stronger forms of choice over the Chang model; the supplied text does not establish the assertion or indicate whether it has been resolved.