Leading-term conjecture for volumes of (1,k)-annuli

Let $V_{\mathbb{A}_{1,k}}$ denote the volume of the moduli space of $(1,k)$-annuli, viewed as a polynomial in $\pi^2$. Here $k$ is a positive integer. Leading-term conjecture for annular volumes. The leading term is

$$\frac{(2k-1)!!}{(2k)!!}\pi^{2k}\log 2$$

when $k$ is odd of the form $2k+1$, and the leading term is

$$\frac{7}{4}\frac{(2k-2)!!}{(2k-1)!!}\pi^{2k-2}\zeta(3)$$

when the annulus has type $(1,2k)$. These formulas are proposed from computed volume values; their validity for all indicated values of $k$ remains open.