Can you make fermented, oil-stored Giardiniera at home?

I'm wanting to try to make Chicago-style giardiniera at home. However, some food-safety site says that

The Italian condiment, Giardiniera, CANNOT be made safely at home. This mixture of raw, hot peppers and vegetables in an oil base can support the growth of Clostridium botulinum. Therefore low-acid foods, such as vegetables, that are mixed with oil must be kept refrigerated. They can be kept refrigerated for up to 10 days. Commercially prepared products contain specific levels of microbial inhibitors or acidifying vegetables for use in an oil-based product in the home environment.

Most recipes I look up for fermented giard are touting it's mystic gut-flora curative powers, and don't discuss storing it under oil. The oil recipes I find just have the vegetables soaked in brine for a day, while refrigerated, so practically nothing microbial is going to happen there.

As to the warnings, I'm not sure if the "low-acid foods [...] that support the growth of Clostridium" include fermented vegetables. From what I can tell, fermented vegetables are more likely to be attacked (in the short-term) by mold, and only if oxygen gets in.

The "proper" Chicago recipe is cited in the Tribune article as broadly:

According to Shay of Local Foods, making Chicago-style giardiniera is a two-step process. "First, you pickle the vegetables," he says. Then, "you drain everything, and then cover (the vegetables) with oil." Shay lets the vegetables pickle for two weeks before tossing them in the oil, where he leaves them to infuse for another two weeks.

My question is basically, what is the expected shelf-life/hazards (refrigerated or not) if I do that (2 week ferment, drain, oil, 2 week infuse).