  In 2015, using an innovative technique, Carando, Defant and Sevilla-Peris
succeeded in proving a Bohnenblust--Hille type inequality with constants of
polynomial growth in $m$ for a certain family of complex $m$-homogeneous
polynomials. In the present paper, using a completely different approach, we
prove that the constants of this inequality are uniformly bounded
irrespectively of the value of $m$.
