Introduction:
It is stated in my book from year 1961 that it is unknown whatever a subsemigroup $D$ of semigroup $S$, which is also a $\mathscr{D}$-class of $S$ is always bisimple, but it is known for $\mathscr{D}$-classes which are regular.
Is it still unknown to this day? If it have been solved, can you provide sources?
Definitions:
Definition 1:
We say that the elements $x$ and $y$ belonging to the semigroup $S$ are in the same $\mathscr{L}$-class ($\mathscr{R}$-class) if and only if they generate the same left (right) principal ideal, that is, if $S^1x = S^1y$ ($xS^1 = yS^1$) holds.
Definition 2:
We say that elements $x$ and $y$ of $S$ belong to the same $\mathscr{D}$-class if and only if $(x, y)\in\mathscr{L}\lor \mathscr{R}$ (the joint of relations $\mathscr{L}$ and $\mathscr{R}$). As an alternative definition, it can be shown that relations $\mathscr{L}$ and $\mathscr{R}$ commute, and hence $\mathscr{D} = \mathscr{L}\circ\mathscr{R} = \mathscr{R}\circ\mathscr{L} $, that is $x$ and $y$ belong to the same $\mathscr{D}$-class if and only if there exists $z$ such that $x\mathscr{L}z\mathscr{R}y$.
Definition 3:
We say that the semigroup $S$ is bisimple, if and only if there is a single $\mathscr{D}$-class in $S$.
Already in 1969 T. E. Hall in a paper “An example involving a non-regular $\mathcal D$-class in a semigroup” answered in the negative (by considering an example) the problem posed in exercise 6 of § 2.3, page 62 of “The Algebraic Theory of Semigroups” by A. H. Clifford and G. B. Preston, namely: If a $\mathcal D$-class $D$ of a semigroup $S$ is a subsemigroup of $S$, then is $D$ necessarily bisimple? The article was communicated by G. B. Preston.