This 2021 review subject was risk relationships between diseases from the perspective of gut microbiota:

“There is a significant inverse relationship between the onset of Alzheimer’s disease/Parkinson’s disease (AD/PD) and cancer, but the mechanism is still unclear. Considering that intestinal flora can connect them, we briefly introduced the relationship among AD/PD, cancer, and intestinal flora, studied metabolites or components of the intestinal flora, and the role of intestinal barriers and intestinal hormones in AD/PD and cancer.

According to existing evidence:

• Bifidobacterium and Lactobacillus positively affect AD/PD and cancer;
• Ruminococcaceae, Prevotellaceae, and Prevotella significantly improve on AD/PD but harm cancer; and
• Blautia has universal anticancer ability, but it may aggravate AD pathology.

This may partially explain the antagonistic relationship between neurodegenerative diseases and cancer. When some individuals suffer from one disease, their intestinal flora change to obtain a stronger resistance to the other disease than healthy individuals, which is consistent with statistical data.”

https://www.sciencedirect.com/science/article/pii/S0753332221011276 “Composition of intestinal flora affects the risk relationship between Alzheimer’s disease/Parkinson’s disease and cancer”