ΔE 20.7, loose: different enough that you should pick on purpose. Dioxazine Purple is a little darker, more saturated, warmer than Dark Sky.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 21). Dioxazine Purple is a little darker, more saturated, warmer than Dark Sky.
The same difference applies the other way round: Dark Sky is a little lighter, more muted, cooler.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Dark Sky on the colour wheel See Dioxazine Purple on the colour wheel
All matches: Dark Sky equivalents · Dioxazine Purple equivalents