That looks like the standard Khaldun Treasure chest setup, in which case it will be random each time (stays same until you open & loot it).
If it works like Khaldun, Lockpicking skill will give you clues. Knowing the colors isn't even really needed; just knowing which of the 8 "gem" boxes you are clicking.
My way of brute forcing it, is the following:
Pick all the first color, for all the slots.
Now, you will set a trap off. Heal, and note the result.
IT will say x/5 are the right color, and y/5 are in the right spot. Since all the ones chosen were the same color, x & y will be the same.
if that number is greater than 0, note it.
Repeat for the next color gem box.
When you get one or more colors that have at least one in the combination, and one color that is NOT, start trying to determine the proper spots for the good color(s).
Example: after testing the first 4 colors on the left side, "diamonds" are 2, "turquiose" are 1, "emeralds" are 1 and "citrine" are 0. (in order, going down the side)
Go through, and put diamond in slot 1, and fill the rest with citrine. If the diamond is in a correct slot, you will get a message saying 1/5 are correct. if not, redo with the diamond in slot 2, with the rest citrine, etc. Typically, you only have to do this 1-4 times with a single gem, as if you don't find the right spot after the 4th, you'll now it goes in slot 5.
Say, in this example, it goes into slot 2, and is a match. Now, you have a second diamond to place. You'll now it doesn't go into slot 1, and that slot 2 is already a diamond. So try it in slot 3, CDDCC. If you get 2/5 right, you're ready to move to the next gem type. if only 1/5, redo, shifting to CDCDC. If it's 2/5 you're set; if it's not, well, it could only go into slot 5, so you're done with that gem. We'll say we've determined a pattern of xDxxD, so far.
Now, moving to the turquoise. Go ahead and put the diamonds in slots 2 & 5, and try the turquoise in slot 1; put citrine in slots 3 & 4. If it is in the right space, it will now say 3/5 are in the correct spots. If not, swap slots 1 & 3. If that doesn't work, you'd then know it goes into slot 4. For this example, we'll say you found it went into slot 3.
This brings you to the green gem (we'll call it emerald, for now). we know the pattern so far is (unknown)/Diamond/Turquoise/(unknown)/Diamond, or x/D/T/x/D.
Try to the green gem in slot 1. If it's not a match, then it will be slot 4. Simple one or the other situation. For this example, it will be slot 4.
That gives us the first unknown, the rest D/T/E/D.
Now, enter the 4 you know, then try the 4 remaining colors, one at a time in the unknown slot (with the 4 already determined in the right spot). When you find the right one, it will automatically pop open.
