...fill up the entire converter?
Yes, as a matter of policy.
Or just enough to get you through whatever your writing task is?
I can only think of two (types of) occasions on which I would have a writing task
that begins and ends the same day (or even within the same week): when I am doing some sort of ink testing/review/comparison, in which case the writing outcome on paper itself is not the goal of the task, or when I was preparing invitations, places cards, thank-you cards or some such for my wedding last year. I'd have to atypically have such reservations about an ink, based on its reputation/notoriety, to expect not to want to write with it at all after testing/reviewing it for normal, everyday or random writing applications.
... I can go for weeks without touching a pen. And then I'd need to change the ink or clean the pen and the ink goes to waste.
Exactly why, as the owner of the object, I always consider a fountain pen — of any make, and not just Lamy — that wouldn't prevent total evaporation and drying out of the contents of a fully-filled converter if capped and unused for three months — to be of inferior quality either in design or in manufacture/execution, even if the brand is Platinum (which I like) instead of Parker (for which I have no time whatsoever).
As always: 1. Implicit in everything and every instance I write on FPN is the invitation for you to judge me as a peer in the community. I think it's only due respect to take each other's written word in online discussion seriously and apply critical judgment. 2. I do not presume to judge for you what is right, correct or valid. If I make a claim, or refute a statement in a thread, and link to references and other information in support, I beseech you to review and consider those, and judge for yourself. I may be wrong. My position or say-so carries no more weight than anyone else's here, and external parties can speak for themselves with what they have published. 3. I endeavour to be frank and truthful in what I write, show or otherwise present when I relate my first-hand experiences that are not independently verifiable. If it is something you can test for yourself and see the results, I entreat you to do so.