I checked it and the data is not available.
Obviously route relations are only propagated to the parent highway, if the parent highway is indexed at a given zoom level.
I will make pull request, to display *cn_network at zoom 12 also.
Below zoom 12, it does not make much sense to me, to display cycle routes, because they will have many gaps, for example cycleway, path and track are not indexed below zoom 12.
But I don't understand, why nameTags="ref" (for *cn_network) seems not to be indexed below zoom 15.
Regards,
Max