These oligonucleotides are relatively expensive and sequencing
experiments even moreso, so I wouldn't try to save money with risky
reagent storage conditions. The liquid volumes are all very small
and you can buy nuclease-free TE by the liter so the cost savings
would be negligible anyway. A surfactant in the storage buffer is
optional, to improve pipetting accuracy, but if you buy a
nuclease-free TE concentrate it's easy to add a little bit of
molecular biology-grade Tween 20 solution when you dilute it into
nuclease-free water in your clean workspace.
IDT has some nice data about oligonucleotide storage:
https://www.idtdna.com/pages/education/decoded/article/storing-oligos-7-things-you-should-know
JWF