It's probably fine for the short term and there's no significant
effect on the reactions, but for long-term storage it's preferable
to store DNA in a buffered solution with a pH around 8 at 25 °C.
Pure water will tend to have a pH below 7 as it reacts with
atmospheric carbon dioxide to form carbonic acid. More information:
https://www.idtdna.com/pages/education/decoded/article/storing-oligos-7-things-you-should-know
EDTA is less essential, since it protects from degradation catalyzed
by divalent cations (the same principle used in this protocol to
fragment RNA), but you probably don't have those in your tubes.
Tween 20 is only there to reduce surface tension and increase
pipetting accuracy, which is not a big deal for this protocol -
that's normally used for qPCR. But I find it easier to make hundreds
of milliliters of the best possible storage solution and use that
for everything rather than maintain a variety of different diluents
for different purposes.
At any rate, the indexed PCR primers in particular are a big
investment and it takes a long time to use them up, so I think if
you spend hundreds of dollars on deluxe custom oligos you can afford
twenty bucks for TE to ensure they're stored safely.