Bailey–Borwein–Broadhurst–Glasser sum rule for Bessel moments

Let $I_0$ and $K_0$ be the modified Bessel functions of zeroth order. For integers $n,k$ satisfying

$$ngeq 2kgeq 2,$$

define

$$Z_{2n,n-2k}:=\sum_{m=0}^{\lfloor n/2\rfloor}(-1)^mbinom{n}{2m}\int_0^\infty [\pi I_0(t)]^{n-2m}[K_0(t)]^{n+2m}t^{n-2k}\operatorname{d}t.$$

B$^3$G sum rule. For every such pair $(n,k)$, the sum vanishes identically:

$$Z_{2n,n-2k}=0.$$

This is an open cancellation formula proposed from high-precision numerical computations for Bessel moments, whose general validity remains unproved.