What's the deal with top bar hive entrance holes

So a longtime dream came true for us recently: we got our first bees!!
A new friend gifted us a topbar hive, and a month later we got our first swarm to fill it with. We had planned to start our beekeeping adventure next year, as currently we are so swamped with all our other work that we didn't have adequate time to read up on everything.
But as the bees basically just fell into our lap, we decided to just dive directly into the deep end and see how it goes, learning along the way.

I'm doing much reading about the ins and outs of a top bar hive, trying to keep up with the actual weather circumstances and the state of the colony. And I've hit a bit of doubt about the flight entrances, and specifically how many i should leave open or closed.

Our hive has 6 entrance holes on the long end of the hive. They can be closed with a wine cork.
The swarm we have seems reasonably strong, we've got both eggs and larva, they are collecting pollen and were starting to build up honey storage. Now we got hit by several weeks of unusually cold and wet weather. Many beekeepers in the area are lamenting that food stores of their bees are running completely out, due to the bees not being able to collect nectar. Everyone is feeding syrup right now, including us.
Warnings have circulated in the bee community that you have to take care to avoid stronger bee families robbing the weaker families blind and killing them in the process. This leads me to the following question:
How many holes do I leave open?
Previously they were using 2 holes, and in warmer hours of the day they were so active that it was already quite the queue at the entrance holes for the bees to get in and out. With the current food shortages and cold temperatures I opted to close the second hole as well, so the hive would be easier to defend against marauding bees. But now as soon as the rain lets up for half an hour, the bees fly out en masse to collect as much food and pollen as possible (their numbers are growing fast!), and with just the one hole open, the entrance has become a veritable bottleneck, bees bumping into one another trying to get in or out, many bees sitting on the sides of the hive waiting to get in.

How many holes should I leave open or closed? What is more important: protecting them from potential invaders during these hard weather circumstances, or allowing them easy acces to the hive so they can collect as much nectar and pollen as possible during those few warmer hours of the day, without them wasting valuable time queuing in front of the entrance to get in or out?

Would there ever be a reason to leave all 6 holes open? Or all closed safe for one?
I find very little advice online on how many holes to leave open relating to the progression of your colony.

Any insight welcome!

my approach has always been to let the bees decide. several of my hives have extra holes in hive bodies because I used knotty wood. the bees seem to close and open these holes with propolis as they see fit.

I'm not sure that I would offer that as advice, because all beekeeping is local, but it has worked for me.

Can't say that I've ever heard of a hive with 6 entrances but it's probably about the same area as a Langstroth hive with no entrance reducer. One or two cork sized holes seems like plenty unless it's a full sized colony. I actually prefer to see a bit of congestion rather than too much open area for them to defend. Try watching the entrance & timing the delay of any given bee. (good luck with that) Rough guess I'd say 30 seconds or less is acceptable. I would also suggest using pure cane sugar water (or whatever natural sugar is available in your area) instead of HFCS. I'd prefer to help them defend what they already have since there's not much outside pollen or nectar available anyway. If other bees get inside they might bring in disease & pests. In my opinion that's much worse than having to supplement their food until conditions improve.

Good luck with your new bees & keep learning as much as you can as fast as you can. Just remember that bees know exactly what they're doing & don't read the same books as we humans do. They usually don't need us much.