>>223180You plant the potato eyes in the bottom then when the potato plants are x height you prune the leaves off that will be in the soil. Let those wounds heal for about 4 days before adding the soil. Repeat the pruning and adding of soil every x amount of new height grown. Once the plant is past the top of the container just let it grow from then on.
I think this is best done in a container that has open sides. Like those made from wire fencing,
https://www.google.com/search?q=wire+fencing+potato+tower&tbm=ischI think this because they breath a bit better and help prevent rot. Making sure the wounds left from the bottom leaf removal are healed before soil is put in is key.
For the potato eyes, make sure you allow them to dry out for 2 days after your cut them out. This also helps prevent them from rotting. Just place them in a cool area out of direct light for that time before planting.
This works for potatoes because they grow new roots and tubers along their stems. In your list, only potatoes do this properly. You really need to be patient with this method. If you rush it, the stems will rot and die.