3 colors have to be adjusted here.
From the target image I took a sample of each, and copy/pasted it on a new layer.
I made a selection for each color that I wanted to adjust. One for Dark green, one for Turquoise, one for Red.
Then I copy/pasted the Dark green on a new layer, the Turquoise on a new layer and the Red on a new layer.
I now adjusted the color separately in each chanel with curves until it showed the same shade as the target sample.
Green chanel before applying curves
Green chanel after applying curves
So for instance if you want to change the Dark green color to the Dark Purple color, you have to adjust the curve in the red chanel, the blue chanel and the green chanel.
The same procedure is necessary for changing the Turquoise color to a Dark Blue color,
and for changing the Red color to a Light purple color.