Not necessarily. Sport or seedling, Loddiges could have obtained commercial rights to it from an amateur-fancier (not necessarily English) who raised/propagated it from a rose in the amateur's collection, the originator perhaps not specifying the identity of that parent rose to Loddiges. But, sport or seedling, Loddiges might indeed have had the parent rose (knowingly or unknowingly) in the Loddiges collection under a name which gives no clue of its possible relation to 'Painted Belgic' ('Leda'). It might be fruitful to scrutinize the various Belgicas/Agathes in the Loddiges collection, and generally, to shed light on the situation. The situation involving the origins of what we now call 'Leda' and 'Pink Leda' is, and has always been, strangely mysterious and intriguing.