  We review recent results and conjectures for a simplified version of the
depinning problem in presence of disorder which was introduced by Derrida and
Retaux in 2014. For this toy model, the depinning transition has been predicted
to be of the Berezinskii--Kosterlitz--Thouless type. Here we discuss under
which integrability conditions this prediction can be proved and how it is
modified otherwise.
