Alzheimer's disease (AD) is a progressive neurodegenerative disease with high inter-patient variance in rate of cognitive decline. AD progression prediction aims to forecast patient cognitive decline and benefits from incorporating multiple neuroimaging modalities. However, existing multimodal models fail to make accurate predictions when many modalities are missing during inference, as is often the case in clinical settings. To increase multimodal model flexibility under high modality missingness, we introduce PerM-MoE, a novel sparse mixture-of-experts method that uses independent routers for each modality in place of the conventional, single router. Using T1-weighted MRI, FLAIR, amyloid beta PET, and tau PET neuroimaging data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), we evaluate PerM-MoE, state-of-the-art Flex-MoE, and unimodal neuroimaging models on predicting two-year change in Clinical Dementia Rating-Sum of Boxes (CDR-SB) scores under varying levels of modality missingness. PerM-MoE outperforms the state of the art in most variations of modality missingness and demonstrates more effective utility of experts than Flex-MoE.