Comment by ABetaMale

The connection is probably that ApoE4 carriers clear out amyloid beta deposits less efficiently than ApoE3/2 carriers (see e.g. [1] among many other sources), whereas microbial infection can induce amyloid deposits to form in the first place (see e.g. [2]), although it's not the only mechanism which can induce such deposits.

[1] Castellano et al (2011). Human apoE Isoforms Differentially Regulate Brain Amyloid-β Peptide Clearance. https://doi.org/10.1126/scitranslmed.3002156

[2] Eimer et al (2018). Alzheimer’s Disease-Associated β-Amyloid Is Rapidly Seeded by Herpesviridae to Protect against Brain Infection. https://doi.org/10.2139/ssrn.3155923