I replaced the regeneration valve, and that solved the ticking problem. For what it's worth, here's the part number of the new valve. Note the part # on the box is different than the part # on the part itself.

A 000 470 55 93 - part # on box of new valve
A 000 416 97 32 - part # on new valve itself
000 470 15 93 - part # on old valve

Now on to solve the next noise...

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1998 E320 120K mi
1991 Miata 80K mi
2001 Ranger 109K mi