Let $\mathrm{BB}:\mathbb{N}\to\mathbb{N}$ be the busy beaver function. Eventually $\mathrm{BB}(n)>n^2$ so the sum$$S=\sum_{n=1}^\infty \frac{1}{\mathrm{BB}(n)}$$converges.

What is the largest rational number $q$ such that $S\geq q$ is not known to be inconsistent with ZFC?

Let $\mathrm{BB}:\mathbb{N}\to\mathbb{N}$ be the busy beaver function. Eventually $\mathrm{BB}(n)>n^2$ so the sum$$S=\sum_{n=1}^\infty \frac{1}{\mathrm{BB}(n)}$$converges.

Question. What is the largest rational number $q$ such that $S\geq q$ is not known to be inconsistent with ZFC?