An explicit upper bound for the Helfgott delta in SL(2,p)

Helfgott proved that there exists a $\delta>0$ such that if $S$ is a symmetric generating subset of $SL(2,p)$ containing 1 then either $S^3=SL(2,p)$ or $|S^3|\geq |S|^{1+\delta}$. It is known that $\delta\geq 1/3024$. Here we show that $\delta\leq(\log_2(7)-1)/6 \approx 0.3012$ and we present evidence suggesting that this might be the true value of $\delta$.