The girth-shifting conjecture for Hall ratios

Let $\rho(d,g)$ denote the maximum Hall ratio among graphs of maximum degree at most $d$ and girth at least $g$. Girth-shifting conjecture. The values presented in the table are upper bounds on $\rho(d,g)$ for

$$d\in\{3,4,5\},\qquad g\in\{6,8,10,12\}.$$

The conjecture suggests that the bounds for girth $2r+2$ coincide with those for girth $2r+3$ in the relevant cases, and would in particular yield an upper bound of $2.5$ on $\rho(3,13)$.