A Sword Nurturing Box was something that was considered almost essential to those that were following the Sword Path.
Sword cultivators would consider their swords as their own body, treating it with the same care. As such, a Sword Nurturing box was something that could be considered as giving the sword a proper place to rest in.
While there was of course, the sheath of a sword, it could be considered more like the 'robes' of a sword.
In the case of Lin Mu, both his immortal swords, Afternoon Pine and Ocean Raker, did not have sheaths. Though this was mostly due to the fact that they were being nurtured in the Deep Sapphire City for a long time.
Even if they had a sheath at one point, they were probably gone.
And since Lin Mu mostly kept the swords in his ring, he didn't need specific sheaths for the swords either.
