Only the first noderef field gets an ASM widget

I've added several noderef fields to my content type and set each of them to ASM Select widget.
Only the first one I created shows the widget: the others show a regular select list (and also the 'create' link).
The settings for the fields are the same (apart from which content types they reference.)

My page source is showing a SCRIPT block in the head for each ASM select widget, and each has its own jquery 2 $(document).ready(function() function. Might this be the problem -- that only the first one gets used by jquery?