MediumRoastSteam I’d keep it empty - in fact, I’d unscrew the filter, shake it as dry as possible and pack it in cling film and store it in the fridge, if you plan to use it only once a week.
It takes about 15 minutes to filter 2 litres of water, so it’s no problem to have it ‘empty’ (BTW, the water will then sit for ‘up to a week’ in your coffee machine, but that is clearly not a problem either, since you seem to be doing that with distilled water today).
Life of the filter is - I understand - totally dependent on water hardness; each filter contains ion-stripping resins for 18,000 mg/CaCO3 equivalent. This may sound a lot, but with water in “our” areas in the South-East we are looking at 40-60 litres/filter.