On the real polynomial Bohnenblust-Hille inequality

It was recently proved by Bayart et al. that the complex polynomial Bohnenblust--Hille inequality is subexponential. We show that, for real scalars, this does no longer hold. Moreover, we show that, if $D_{\mathbb{R},m}$ stands for the real Bohnenblust--Hille constant for $m$-homogeneous polynomials, then $\displaystyle\lim\sup_{m}D_{\mathbb{R},m}^{1/m}=2.$