It is more about the "duration of treatment" with prednisone rather than the dose. If a low dose is 5 mg or less for a short period of time, maybe the risk is lower. However, 5 mg for a longer period of time, the risk of adrenal insufficiency may increase.

The following abstract addresses the risk of long term low dose glucocorticoid treatment.
https://www.endocrine-abstracts.org/ea/0037/ea0037ep46
I like this abstract because it mentions the "mismatch" between glucocorticoid requirements and production/intake. I'm thinking the production is the cortisol that the adrenals produce and the intake is the prednisone taken orally. The total daily requirement is never a fixed amount and depends on the day. I usually took a fixed amount of Prednisone on any given day but my requirement might increase. If my requirement increased and my adrenals weren't producing cortisol, I think that would create a mismatch. Perhaps this explains why many people experience flares during stressful situations at lower doses of prednisone.

I was able to taper down to 3 mg of prednisone because Actemra made it less likely I would have a PMR flare. It took me a while to convince myself that a PMR flare wasn't going to happen. At this stage, my rheumatologist stepped aside and referred me to an endocrinologist when my cortisol level was low. I wasn't having any PMR pain but I can't say I felt well because of the overwhelming fatigue.

I was instructed by my endocrinologist to stay on 3 mg of prednisone for an extended period of time. The 3 mg dose was a very low dose for me. My endocrinologist was surprised that I had a measurable cortisol level given I was on high doses of prednisone for a long time. I didn't feel well on the 3 mg dose but my endocrinologist "encouraged me" to stay on that dose. She also said I could increase my dose "if I felt the need."

My endocrinologist rechecked my cortisol level after 6 months and was surprised it was in the low normal range. She described my cortisol level as "adequate." She wasn't sure what would happen if I discontinued prednisone. When I decided to discontinue prednisone she provided me with a direct number to call. She said preferably I should call before taking prednisone again but didn't require me to call first. She even stressed once again that I should take Prednisone again if I felt the need.

My first attempt to discontinue prednisone didn't go so well but that's another story. What happened was completely unexpected but I needed 60 mg of prednisone again and an ophthalmologist got involved. Fortunately, it was only a temporary setback.

This is another article that may speak to low dose prednisone for a short duration.

"Conclusions
The heterogeneity of studies and variability in results make it difficult to answer the research questions with confidence based on the current literature. There is evidence of AI following low doses and short durations of glucocorticoids. Hence, clinicians should be vigilant for adrenal insufficiency at all degrees of glucocorticoid exposure."
https://www.sciencedirect.com/science/article/pii/S0049017216000913