The site labels the problem PROVED, but its commentary records the answer of Hegyvári, Hennecart and Plagne [HHP07]: yes for $k=2$, with $b_{n+1}-b_n\le2$ for all large $n$, and no for every $k\ge3$. The Statement asks whether the gaps are bounded for every basis of every order $k$. Erdős's own wording, quoted in the introduction of [HHP07] from [Er98], asks the same for general $k$ ("Is it true that $\limsup(b_{i+1}-b_i)<\infty$ ... The bound may of course depend on $k$ and on the sequence"). Theorem 1(ii) of [HHP07] gives, for each $h\ge3$, a set $A$ with $h(\{0\}\cup A)$ containing all large integers, a basis of order $h$ in the problem's sense, whose sums of $h$ or fewer distinct elements have unbounded gaps; the authors describe the result as "a negative answer to a question by Burr and Erdős" and "an explicit counterexample to the Erdős-Burr conjecture". So the Statement is disproved, and the case $k=2$, where Theorem 1(i) gives $b_{n+1}-b_n\le2$ for all large $n$, is the part of the question that holds (Hegyvári, Hennecart and Plagne, accepted, full). The page departs from the site's label here: PROVED, the site's "solved in the affirmative", contradicts both the theorem in print and the site's own commentary, and no source offers a reading of the question under which the answer is yes.