Shrikhande and Singhi [ShSi85] have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power, by proving that every pairwise balanced design on $n$ points in which each block is of size $\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\leq c+2$, if $n$ is sufficiently large.

Erdős asks if this is false for constant, for which functions $h(n)$ will the condition $\lvert A_i\rvert \geq n^{1/2}-h(n)$ make the conjecture true?