Data from the GENOA study, which began in 1997 and involved African Americans with high blood pressure and their siblings, has found that obesity as such was not linked to cognitive decline. The key factor was abdominal fat rather than BMI.

Having too much belly fat in mid- and late-life was linked to a much higher risk of cognitive decline. Among middle-aged adults, losing weight was linked to higher cognitive scores, while gaining weight was linked to lower scores. However, in later life, losing weight was linked to lower scores (I note that losing weight in old age is often a marker of developing dementia), and gaining weight was linked to better scores.

People in the study ranged in age from 35- to 86-years-old; 78% were women and most were obese. Participants were assessed in 1997-99, 2001-6, and 2009-11.

https://www.eurekalert.org/pub_releases/2017-05/ags-teo050817.php

West, N. A., Lirette, S. T., Cannon, V. A., Turner, S. T., Mosley, T. H., & Windham, B. G. (2017). Adiposity, Change in Adiposity, and Cognitive Decline in Mid- and Late Life. Journal of the American Geriatrics Society, 65(6), 1282–1288. https://doi.org/10.1111/jgs.14786