The formulation of the mean-field infinite-dimensional solution of hard sphere glasses is a significant milestone for theoretical physics. How relevant this description might be for understanding low-dimensional glass-forming liquids, however, remains unclear. These liquids indeed exhibit a complex interplay between structure and dynamics, and the importance of this interplay might only slowly diminish as dimension d increases. A careful numerical assessment of the matter has long been hindered by the exponential increase in computational costs with d. By revisiting a once common simulation technique involving the use of periodic boundary conditions modeled on D_d lattices, we here partly sidestep this difficulty, thus allowing the study of hard sphere liquids up to d = 13. Parallel efforts by Mangeat and Zamponi [Phys. Rev. E 93, 012609 (2016)] have expanded the mean-field description of glasses to finite d by leveraging the standard liquid-state theory and, thus, help bridge the gap from the other direction. The relatively smooth evolution of both the structure and dynamics across the d gap allows us to relate the two approaches and to identify some of the missing features that a finite-d theory of glasses might hope to include to achieve near quantitative agreement.