"Diamond" fill effect displaying successive rectangles instead of starburst effect

[Advocate]

I'm creating cards for different decks. What the printer gets and what I get are two totally different things. This card (.xar file attached), has been designed with the diamond fill. I got poor results when I exported the .xar file itself to PDF format.

So I exported the .xar file as a 300dpi png, and next exported the PNG file to PDF, using Xara DP6 program. Turned out exactly as I want it. File attached for anyone who wants to see.

So, I send the PNG file to the printer. He exports it to Adobe Illustrator, and it comes back with rectangles instead of the subtle starburst effect. I have four card designs that use this diamond effect, and they all do the same thing. His PDF file is too large to attach here because there are 11 card samples on it and it's 860kb. So I'm attaching a screenshot of how the red card turned out when he exported the PNG file to PDF. I'm also attaching a screenshot of the blue card, where you see the rectangles are quite prominently displayed.

(Sorry about the blue card image being so large; I was working with Photoshop Elements and must have forgotten to resize.)

Two questions:
1 - Does anyone know what the heck the problem is?
2 - I told him I could export the cards to PDF and send them instead of PNGs, but haven't heard back. Does anyone have experience with taking a PDF export from Xara and using it in an Adobe Illustrator template?

I hope that all makes sense. Any helpful advice is most welcome. I'll check back later tonight. TIA