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 ${\displaystyle \lim \underset{m}{\sup }{D}_{\mathbb{R},m}^{1\mathrm{/}m}=2.}$