To answer the question, I'll put some bounds to the conditions for a more accurate view ...
- Assuming the two oils (conv and syn) are reasonably modern and not "old stock" from decades ago.
- Assuming the engine application is not a known sludger, or a shear-monster, has known mechanical flaws, etc.
- Assuming the conditions of operation are "normal" and not uber extreme (below 0 degF; drag racing cycles in AZ; etc)
- Assuming the maintenance plan is "normal" 3k miles with a decent filter (can often go further than 3k miles, but that is the OPs quest here)
- Assuming we're not going to nit-pick the word "synthetic" and just accept that it could refer to a grp III, IV or V. (a worthy topic for a different day)
Generally, the answer would be "yes", a conventional oil will provide the same wear protection as a syn.
Wear protection comes from three things:
1) tribo-chemical barrier
2) MOFT
3) additive package
I cannot state which of those three are most important; I believe they all three contribute to an overall measure of success, and to the best of my knowledge, there has never been an SAE or other credible study that has isolated one relative to the others for a determination. But rest assured, the three attributes combine to make for wear control.
I have personally run OCIs out to 10k miles in 4.6L Ford engines, and seen absoblutely zero wear differences between conv and syns:
https://bobistheoilguy.com/forums/threads/5w-20-ford-4-6l-engines-uoa-testing.269435/page-3
That does not mean it will always be true; it was true in this case because the assumed conditions I stated were true. This is just real-world annecdotal proof that the claim is real, if the conditions are right. (NOTE: Individual UOAs will always have some variability; always use statistical data norms for your comparison and contrast limits.)
I also have around 30k UOAs in my database; so many now that I've stopped collecting them. I can say with absolute certainty that if the assumed conditions are true, the answer is an emphatic "Yes" - conventional oils will protect as well as syns. I've studied these UOAs until my eyeballs ached, and I can say unequivically that if the assumed conditions are in place, you would never be able to tell a syn from a conventional oil in terms of wear control.
- Syns are superior at holding their vis in range, but that ONLY is a benefit if the conventional oil has been used long enough that it goes severly out of range; probably never going to happen in a 3k mile OCI.
- Syns are superior at cold start pumpability, but that NEVER is a benefit if it's not uber cold outside.
- Syns are superior at oxdiation control, but that NEVER is a benefit if the engine does not overheat, because "normal" expected operational temps are not a problem for conventional oils. (Note: some oxidation is good; it's what builds the TCB layers; SAE 2007-01-4133)
Further, syn lubes are not immune to severe conditions either. Yes, syns can endure some conditions better than a conventional oil, but you absolutely run a syn oil for too long of an OCI. A syn most certainly can shear. A syn can degrade with heat. A syn can have it's additive package overwhelmed with contamination and degradation just like any conventional oil. Typically, it just takes "more" of the offensive condition to make a syn fail relative to where a conventional would fail.
Syn lubes
ONLY have an advantage over a conventional oil
IF some condition experienced makes the convetional oil fail first. Given the stated assumptions, that would never be true, and therefore the logical and truthful answer is that you'd never see a difference in wear control or even cleanliness for that matter.