Multilinear Marcinkiewicz integral extrapolation conjecture

Let $\Omega$ be a function on $({\mathbb R}^n)^m$ satisfying conditions (i), (ii), and (iii) in the definition of a multilinear Marcinkiewicz integral. For Schwartz functions $\vec f=(f_1,\ldots,f_m)$, define

$$F_t(\vec f)(x)=\frac{1}{t^m}\int_{(B(0,t))^m}\frac{\Omega(\vec y)}{\prod_{i=1}^m|y_i|^{n-1}}\prod_{i=1}^m f_i(x-y_i)\,d\vec y,$$

and

$$\mu(\vec f)(x)=\left(\int_0^\infty|F_t(\vec f)(x)|^2\,\frac{dt}{t}\right)^{1/2}.$$

Multilinear Marcinkiewicz integral extrapolation conjecture. If $\mu$ is bounded from $L^{q_1}({\mathbb R}^n)\times\cdots\times L^{q_m}({\mathbb R}^n)$ to $L^q({\mathbb R}^n)$ for some $1<q_1,\ldots,q_m<\infty$ satisfying $1/q=1/q_1+\cdots+1/q_m$, then $\mu$ is bounded from $L^{p_1}({\mathbb R}^n)\times\cdots\times L^{p_m}({\mathbb R}^n)$ to $L^p({\mathbb R}^n)$ for every $1<p_1,\ldots,p_m<\infty$ satisfying $1/p=1/p_1+\cdots+1/p_m$. This conjectures that boundedness at one admissible tuple of exponents implies boundedness at every admissible tuple, under the stated kernel conditions.