Multiplication is an open bilinear mapping in the Banach algebra of functions of bounded Wiener $p$-variation

Let $BV_p[0,1]$, $1\le p<\infty$, be the Banach algebra of functions of bounded $p$-variation in the sense of Wiener. Recently, Kowalczyk and Turowska \cite{KT19} proved that the multiplication in $BV_1[0,1]$ is an open bilinear mapping. We extend this result for all values of $p\in[1,\infty)$.