Context. The advent of ubiquitous large-scale distributed systems advocates that tolerance to various kinds of faults and hazards must be included from the very early design of such systems. Self-stabilization [1] is a versatile technique that permits forward recovery from any kind of transient fault, while Fault-tolerance [2] is traditionally used to mask the effect of a limited number of permanent faults. The seminal works of [3,4] define FTSS protocols as protocols that are both Fault Tolerant and Self-Stabilizing, i.e. able to tolerate a few crash faults as well as arbitrary initial memory corruption. In [3], some impossibility results in asynchronous systems are presented. In [4], a general transformer is presented for synchronous systems. The transformer of [4] was proved impossible to transpose to asynchronous systems in [5] due to the impossibility of tight synchronization in the FTSS context. It turns out that FTSS possibility results in fully asynchronous systems known to date are restricted to static tasks, i.e. tasks that require eventual convergence to some global fixed point (tasks such as naming or vertex coloring fall in this category).