Boundary cross theorem in dimension 1 with singularities

Let $D$ and $G$ be copies of the open unit disc in $\text{C},$ let $A$ (resp. $B$) be a measurable subset of $\mathrm{\partial }D$ (resp. $\mathrm{\partial }G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$andlet$M$bearelativelyclosedsubsetof$W.$Supposeinadditionthat$A$and$B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\mathrm{\varnothing }\mathrm{.}$ We determine the “envelope of holomorphy” $\widehat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\setminus M,$ measurable on $A\times B,$ and separately holomorphic on $((A\times G)\cup (D\times B))\setminus M$ “extends” to a function holomorphic on $\widehat{W\setminus M}\mathrm{.}$