I think greedy works. Sort tasks by start time. Sweep over the time units. At each point we have a choice — Take this time unit or wait and take a later one. If we can take a later time and still complete the task with the smallest difference (time left in interval — time units still needed), it's better to choose the later time. Reason is that choosing the later time still guarantees that we complete that task, but going to a later time might mean that we step over a new interval start and share the chosen unit with more tasks. We can never share with less tasks by taking a later time because if some task completes before this one we are guaranteed to share all their remaining time units.

"But wait, that takes 1e3*1e6=1e9 and is too slow!?" Not if we go by intervals — There are only 3*1e3 interesting times total. Just step to each of them and do math to see if/how many units we need to choose on each interval. Also consider keeping active intervals in a heap by (time remaining — time still required).