This is one of the hardest spat to remove but there is a way.
Be very care full using same products because you may expend the spat rather than remove it but try this it should work even if it Sound strange.
In a cup of warm water add a drop or two of dish soap and same ice cube and mix it.
With a piece of cotton deep it in the cup squeeze out exes water and rub on stain. Every now and than take the ice and rub on stain also.
This should take the ink out, I hope it will help.
Good luck.

“Sard” soap,warm water and a “nail bush”. You can find it near the soap powders in the store. It would have been easier to use believe it or not “Ketchup” first would have removed it. Since you used rubbing alcohol and club soda you have embedded the stain. Soak the stain with a wet sponge sprinkle the stain with sard soap. (If they only sell it in blocks rub the soap into the stain,) leave it and start on another stain. Return to the first stain lightly agitate with a nail brush dipping it into the water. Using the sponge wipe the pink foam and stain. It’s only the oil from the ink that still is embedded in your carpet. “repeat this process on all the stains” By using a small nail brush it won’t spread the stain or wear the carpet. Depending how deep the stains are you might have to do it again. This should remove the remove the ink stains.

have you tried acetone, then brush it with soap in warnm water, try it, done that, and it works.

You could try shaving cream. The foamy kind. It works on stains. It depends on how worked in the stain in now after trying the other things.

there is a soap called ZOTE it comes in a bar form I have had alot of luck getting all kinds of stains useing this it is worth a try good luck hope it helps