Bohnenblust--Hille inequality for polynomials whose monomials have uniformly bounded number of variables

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 $m$.