Printing of the colors do not match proof on the monitor

I received my printed product and the colors do not match what I saw on my screen. 

Here's a question we received from a customer who placed an order with this same issue. 

Problem 1. Dark Black colors inaccurate to digital proof

The questions was: 

"The printing on the items is very strange. It looks like the colors did not come out correctly and does not match the proof. Please let me know if there is anything we can do to remedy this. Thanks!"

On screen, here's how my design appear: 



Here's how it looks side by side with the actual printed piece. 




Reason why this happened: 

The file provided by the customer is not of suitable quality for printing and because of this the print quality does not look as the customer expects! The digital machines we use are very precise and print even very small details that are not visible to the eye.

The background of the product is a photo and has a variable percentage of one color black from 97% to 100% at the edge of the picture and around the text on the background there is something like a shadow (it looks slightly in the dark purple range), which is four colors C-13%, M-13%, Y-13% and K-98%. If you look carefully the attached photos, you will notice it with your eyes. 

Resolution:

Option 1:

Provided by the customer high resolution vector file

Option 2:

Convert the file to grayscale. This will enable the digital machine to reproduce the color/ colors of picture smooth( with all gradations of black) to an acceptable standard.

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