Construct pairwise balanced designs with $O(\sqrt{n})$ blocks of every size (Erdős #734)

A collection $A_1,\ldots,A_m\subseteq\{1,\ldots,n\}$ is a pairwise balanced block design if every pair of distinct elements of $\{1,\ldots,n\}$ is contained in exactly one $A_i$. Find, for all large $n$, a non-trivial pairwise balanced block design on $\{1,\ldots,n\}$ such that, for every $t$, the number of indices $i$ with $\lvert A_i\rvert = t$ is $O(n^{1/2})$. (Non-trivial excludes the design consisting of the single block $\{1,\ldots,n\}$; the implied constant must be uniform in $n$ and $t$.)

Acceptance. FULLY RESOLVES: an explicit construction scheme, with complete proof, producing for every sufficiently large $n$ a non-trivial pairwise balanced design on $\{1,\ldots,n\}$ in which every block size occurs $O(n^{1/2})$ times (explicit constant stated); OR a proof that no such designs exist for infinitely many $n$ (refuting Erdős's expectation). Machine-checkable proof preferred, else full written proof. ADVANCES: such a construction proved for an infinite family of $n$ (e.g. $n$ of a special arithmetic or prime-power-derived form); a construction achieving $O(n^{1/2+\epsilon})$ or $O(n^{1/2}\log n)$ multiplicities for all large $n$, with proof; a sharpened unconditional lower bound improving the $\gg n^{1/2}$ multiplicity obstruction stated in the background; or explicit verified designs meeting the $O(\sqrt{n})$ target (stated constant) for every $n$ in a substantial contiguous range, delivered as data files plus an independent checker program that verifies the pair-cover and per-size counts. Deliver the construction + proof, or the design files + checker code + the verified range.