Arithmetic nature conjectures for odd Bessel problems

Let $k$ be an odd integer, and let $G_{2k+2}^{\mathrm{BR}}$ be a Gröbner basis generating the Broadhurst–Roberts ideal

$$I^{\mathrm{BR}}_{2k+2}:=\left\langle {\mathbf X}_k\mathfrak d_k{\mathbf X}_k^{\mathrm T}-\mathfrak b_k\right\rangle.$$

Let $\mathbf N_k$ be the matrix of on-shell Bessel moments. Arithmetic nature conjectures. (a) If $k>1$, every linear polynomial in $G_{2k+2}^{\mathrm{BR}}$ is a consequence of the relations for generalized Crandall numbers. (b) If $k>3$, the $(k-1)k$ elements in all rows except the bottom row of $\mathbf N_k$ are linearly independent over $\mathbb Q$. These conjectures concern the arithmetic and linear-algebraic structure of odd Bessel problems; the supplied text presents them as empirically motivated and gives no resolution.