I found a counter-example to the following conjecture.

**Conjecture** $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$.

The counterexample is $latex f = {(=)}|_{\Omega}$ where $latex \Omega$ is the Fréchet filter.

See Algebraic General Topology and in particular Funcoids and Reloids online article, the section Some counter-examples

for this counterexample.