Not only you always need at least a resistor, but matching the power supply voltage to the leds drop will ensure you’ll have a hard time maintaining their brightness on the real world.
If you want to keep it simple, leave 1V or 2V as a margin and add a resistor to get the desired current. If you want to make it fancy, get a led driver.
For example, as the leds on the original circuit age, you can expect a few of those lines to stop getting any current, and the others to get brighter and brighter until they fail.
That can happen very quickly too, if their temperature varies a lot (how much is a lot depends on the leds). Or if you change your power supply for one capable of more current, it’s likely that everything will just fail immediately. Or, if you have a noisy one, you may severely reduce the leds lifetime.
As a rule, you can’t use diodes internal resistance as a power limiter. It does not behave the same way as resistors and will surprise you on every bad way possible.