One of two things happened: either a) you did not compact the soil well enough around the stem, or b) the plant got the damping off fungus. If a), just repot it all over and things will be fine. But if b), you're better off at this point to start over.
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Do a search to get more info on damping off. Basically, it's a soil-borne fungus that mostly young plants are likely to get. If the plant looks weak or watery at the soil line, it's damping off. Your plant is still too young to recover from it, which is why I suggest starting over. That said, the key thing is to keep the surface of the soil dry. This can be accomplished by a) bottom-watering, ensuring the water is not allowed to reach the soil surface, and b) keeping a fan blowing in the room with the plants. The fan will also help strengthen the stems, which will help reduce the time it takes to harden off the plants when you're ready to start bringing them outside.Â