How to remove white background from transparent Illustrator graphic?

To get the publication to print accurately, I did a work-around to get the below-mentioned graphic to print without a white background. But I would like to know how to prevent/fix this in the future & not have to sacrifice the image's transparency/layers. This graphic is part of a larger printed piece, client submitted packaged InDesign file. All links' (Illustrator & PhotoShop) transparency backgrounds work when the publication pages are exported from InDesign - except this one: this AI graphic is text with gaussian blur and outer glow applied. (This graphic, along with a number of others on the printed piece, are exported out of InDesign as separate files to maintain page position for variable fields during printing.) This is the only graphic within the publication that exports with solid white where it should be transparent. I checked the AI document for transparency, I tried options of saving AI file as pdf, eps, and svg, exporting as png: they all placed in InDesign as visually transparent. But when the InDesign exported pdf (PDF/X-4) of this graphic is imported into the variable data end program the transparency is gone; this end program shows how things will print. I have attached pics of the AI graphic & its Transparency Preview, the InDesign view & its Transparency Preview, and the end-program view showing the appearance of white.