Advances in surgical and medical management have improved survival for individuals with Fontan physiology (FP), increasing attention to long-term neurocognitive impairments. We propose the Cardiopulmonary–Brain Axis Hypothesis, identifying respiratory muscle performance (RMP) as a potentially modifiable contributor to executive function in FP. Drawing on evidence from cardiovascular and respiratory physiology, autonomic neuroscience, cerebrovascular biology, and neurocognitive research, this framework explores potential relationships among RMP, FP, respiratory muscle metaboreflex activation, autonomic regulation, cerebral perfusion, and executive function. Fontan-specific studies demonstrate the influence of respiration on venous return and pulmonary blood flow, while selected inspiratory muscle training (IMT) studies report improvements in respiratory function. Building on these findings, we hypothesize that improvements in RMP following IMT may enhance respiratory pump efficiency, favorably influence Fontan hemodynamics, attenuate respiratory muscle metaboreflex activation, and support autonomic regulation and cerebral perfusion, with potential downstream improvements in executive function. However, evidence supporting these relationships varies, with several mechanisms derived primarily from non-Fontan populations. The pathway remains untested, and improvements in executive function following IMT have not been established in FP. The Cardiopulmonary–Brain Axis Hypothesis provides a biologically plausible, testable framework to guide future longitudinal and interventional investigations examining the potential relationships between RMP, cardiopulmonary function, and brain health.