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Claim. Let $A\subseteq\mathbb{N}$ be an asymptotic basis of order $2$: every large integer is a sum of at most two elements of $A$. Hennecart proves that $A$ has a restricted order, in the sense of Problem 338, and that it is at most $4$: every large integer is a sum of at most four distinct elements of $A$. He also constructs a basis of order $2$ whose restricted order is exactly $4$, which refutes Kelly's conjecture that $3$ always suffices. The paper is Hennecart, François, On the restricted order of asymptotic bases of order two, Ramanujan J. 9 (2005), no. 1--2, 123--130. Its summary states the theorem as "We show that any asymptotic basis of order 2 has a restricted order at most equal to 4". Hegyvári, Hennecart and Plagne [HHP07] (library card) record that this settles the case $h=2$, with $f(2)=4$ for the largest restricted order of a basis of order $2$. The theorem extends Kelly's bound $4$ for classical bases of order $2$, recorded at Kelly 1957, to all asymptotic bases of order $2$; the site's remarks state the asymptotic bound and credit it to Kelly.

Covers. Asymptotic bases of order $2$. For them the statement's first question is settled, since a restricted order always exists and no further condition is needed, and its second question is settled with the bound $4$, which the paper's example shows is best possible. The claim's value is answered because the result determines what the first question asks to determine for this class and gives the best bound for the second. Nothing is covered for bases of order $3$ or more, nor for the statement's third question, the conditions under which the restricted order equals the order.

Acceptance. The refereed evidence is the journal publication cited above, in the Ramanujan Journal. The site labels the problem OPEN, so its remarks credit the paper without settling the problem and no reviewed evidence is listed. The publication record dates the issue to March 2005 and gives no finer date, so the page is dated to the first day of that month.

Depends on. Nothing in this wiki.