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Alzheimer's Disease (AD) is the most common form of dementia and often difficult to diagnose due to the multifactorial etiology of dementia. Recent works on neuroimaging-based computer-aided diagnosis with deep neural networks (DNNs) showed that fusing structural magnetic resonance images (sMRI) and fluorodeoxyglucose positron emission tomography (FDG-PET) leads to improved accuracy in a study population of healthy controls and subjects with AD. However, this result conflicts with the established clinical knowledge that FDG-PET better captures AD-specific pathologies than sMRI. Therefore, we propose a framework for the systematic evaluation of multi-modal DNNs and critically re-evaluate single- and multi-modal DNNs based on FDG-PET and sMRI for binary healthy vs. AD, and three-way healthy/mild cognitive impairment/AD classification. Our experiments demonstrate that a single-modality network using FDG-PET performs better than MRI (accuracy 0.91 vs 0.87) and does not show improvement when combined. This conforms with the established clinical knowledge on AD biomarkers, but raises questions about the true benefit of multi-modal DNNs. We argue that future work on multi-modal fusion should systematically assess the contribution of individual modalities following our proposed evaluation framework. Finally, we encourage the community to go beyond healthy vs. AD classification and focus on differential diagnosis of dementia, where fusing multi-modal image information conforms with a clinical need.