Open, the site's label (OPEN; page last edited 2026-04-15). No claim settles the question for every infinite support, so the frontmatter standing, open/none, follows. Accepted partial claim pages record the settled classes of supports: Erdős's 1948 theorem for $A=\mathbb N$ and its sets of multiples, Erdős's 1968 theorem for pairwise coprime supports with convergent reciprocal sum, Duverney and Tachiya's theorem for the sets $F_s(E)$ of products of powers below $s$ of a pairwise coprime, polynomially bounded sequence, such as the squarefree integers, and Tao and Teräväinen's theorem for the primes. One pending partial claim, Cook's 2026 note, asserts irrationality for every support with convergent reciprocal sum.