  We study maximally supersymmetric irrelevant deformations of the D1-D5 CFT
that correspond to following the attractor flow in reverse in the dual half-BPS
black string solutions of type IIB supergravity on K3. When a single, quadratic
condition is imposed on the parameters of the 22 such irrelevant deformations,
the asymptotics of the solution degenerate to a linear dilaton like spacetime.
We identify each such degeneration limit with a known decoupling limit of
string theory, which yields little string theory or deformations thereof (the
so-called open brane LST, or ODp theories), compactified to two dimensions.
This suggests that a 21-parameter family of the above deformations leads to
UV-complete theories, which are string theories decoupled from gravity that are
continuously connected to each other. All these theories have been argued to
display Hagedorn behaviour; we show that including the F1 strings leads to an
additional Cardy term. The resulting entropy formula closely resembles that of
single-trace $T\bar T$-deformed CFTs, whose generalisations could provide
possibly tractable effective two-dimensional descriptions of the above web of
theories.
  We also consider the asymptotically flat black strings. At fixed temperature,
the partition function is dominated by thermodynamically stable, small black
string solutions, similar to the ones in the decoupled backgrounds. We show
that certain asymptotic symmetries of these black strings bear a striking
resemblance with the state-dependent symmetries of single-trace $T\bar T$, and
break down precisely when the background solution reaches the large black
string threshold. This suggests that small, asymptotically flat black strings
may also admit a $T\bar T$ - like effective description.
