The finite-field product-set arithmetic progression conjecture

Let $p$ be a prime and let $B\subset\mathbb{F}_p$ be a set, with $p$ and $|B|$ large. Write $B.B=\{bb':b,b'\in B\}$, and let an arithmetic progression (AP) in $B.B$ mean a set of the form $\{a,a+d,\ldots,a+(L-1)d\}$ with $d\ne 0$.

Finite-field product-set conjecture. There is an absolute constant $c>0$ such that, whenever

$$|B|<c\sqrt p,$$

the product set $B.B$ cannot contain an arithmetic progression of size greater than $|B|^{1+o(1)}$.

Sets $B$ of size $O(\sqrt p)$ can have $B.B=\mathbb{F}_p$, so the restriction to smaller sets is essential. The conjecture proposes an essentially linear upper bound, up to the factor $|B|^{o(1)}$, for the length of an arithmetic progression in a finite-field product set.