Do you have this in your possession, or is this a picture your customer sent to you?
You're right that the linen is more of the issue than wool. I worry about the dark black against the white in the pattern. I've seen some rugs nearly identical to this one in both style and fiber content, and the black dye was unstable to the products typically used in area rug cleaning. The mild reducers suitable for wine also might cause colorloss on the black dye, so test that carefully. You won't see the test result as quickly as the bleed test, you'll need to let the test material dry on it to be sure.