A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the product set $B.B$ is used. Now suppose that instead of $B.B$ we have a partial productset $B\stackrel{G}{.}B$ along the edges of a graph $G$ of edge density $\epsilon$ (meaning that $b_1b_2 \in B\stackrel{G}{.}B$ if only if $b_1, b_2 \in B$ are adjacent in $G$).

Is it true that if $B < \sqrt{p}$ a similar estimate
$$
 |B\stackrel{G}{.}B + B\stackrel{G}{.}B + B\stackrel{G}{.}B - B\stackrel{G}{.}B - B\stackrel{G}{.}B - B\stackrel{G}{.}B| \gg_{\epsilon} |B|^2 
$$
holds?

**UPD**
Sorry for sloppy notation. So what it means. $B.B = \{b_ib_j |b_i, b_j \in B \}$, 
$B.B-B.B = \{b_ib_j - b_kb_l |b_i, b_j, b_k, b_k \in B\}$, and $B.B+B.B$ etc. being defined the same way.