Herb, my experience with Spectraply is limited to pens and handles for bottle openers and coffee scoops.
Here's my short answer: stabilizing Spectraply isn't necessary and has little benefit.
I came to that conclusion after performing a highly subjective test. Regrettably, I don't have pictures to illustrate.
I chose 2 x 14 different colors of Spectraply (i.e., 28 blanks). I stabilized one set of the colors, and left the other set of colors "raw". Then I made 28 pens (slimline or trimline). I used the same component type and plating for each color pair, so that type and plating wouldn't interfere with a visual comparison.
I used a skew for all pens, sanded to final fit using the same process, and finished all the same (walnut oil after final sand, CA finish, micromesh through all grits, buffed before assembly.
Observations (again, subjective)
-Stabilizing caused darker colors (blacks, blues, purples) to bleed, which appeared to slightly muddy brighter colors (yellows, light greens, whites, and no dye). It wasn't a severe change, but I could detect it on the margins of the layers.
-I observed more what I'm calling "micro tear out/fuzziness with stabilizing than without. That surprised me, because I expected the opposite. Even with a sharp skew, this meant for stabilized Spectraply I had to turn more proud of final diameter and sand more to final diameter.
-I didn't have any problems with either stabilized, or non-stabilized delaminating when a layer approached micro-thin.
-Non-stabilized had more sanding dust stick in the pores, but a puffer bulb and liberal use of denatured alcohol took care of that.
-After final sanding, applying oil, CA finish, final sanding, and buffing, non-stabilized looks brighter and margins of the layers were more sharply defined (not muted/muddied/color bled)
-After about 6 months of not looking at the color pairs, I pulled them out to see whether I could still tell a difference. I can.
-Stabilizing risks color bleed of dark into light/bright. It's not severe, but it's enough to notice.
-I won't stabilize Spectraply going forward. I think it's neither helpful, nor harmful. It adds unnecessary steps and cost (time and stabilizing resin), and yields a turned item that the casual observer wouldn't notice is different from a non-stabilized item.