Air Filter Quality vs. DPF Lifespan

Something occurred to me as I was driving home from a friends house last night during one of our horrible west Texas dust storms: Can the filtering abilities of the air filter for the engine intake air affect DPF life? Let me explain:

So, some of us may know that DPFs filter the exhaust of our diesel cars/trucks due to federal emissions laws mandating them. DPFs are designed to catch primarily soot (carbon particles) and regenerate it out using high EGTs for a period of time. DPFs also collect ash, which comes from "the other crap" that isn't just carbon particulates, and ash stays and slowly builds in the DPF until the DPF is removed and thoroughly cleaned or just replaced. This is considered the death of the DPF, and is typically very expensive.

So, where do the engine the air filters come into play here? Well, I'm not 100% sure, but I believe our factory air filters for the 2.0L CR TDI won't filter the air going into the engine as finely as the DPF filters the exhaust coming out. So, does that mean that all of the super fine particles of dust and dirt in our air that get pulled through the engine air filter get run though the engine and then caught in the DPF? Since these dust particles are foreign substances and are not carbon soot particles that can be regenerated away, do they contribute noticeably to ash levels in the DPF? Does this mean that people running intakes with oiled filters which we all know do not filter the air as well as the stock filters have a shortened DPF life due to quicker ash buildup in the DPF? Will people who operate their vehicles in dusty conditions experience reduced DPF life?

I'm genuinely curious.

Opinions?

Anything bypassing the filter that is combustible will be ultimately turned into ash or end up in the oil pan. Some portion of the ash may remain in the DPF. Anthing that is not combustible would either end up in the oil, pass through the dpf or get stuck in it.

So technically there would be some contribution to DPF fouling by particles which bypass the air filter, but I would think that this fraction would be extremely miniscule, even in comparison to the fouling caused by ash from the engine oil, which is also fairly miniscule.

One of my customers had a "cold air intake" on his car and had the unfortunate experience of the air filter falling off the tube without him knowing. His DPF was fine. Engine, not so much....