Multilinear extrapolation conjecture for Banach function spaces

Let $p_1,\ldots,p_m\in(1,\infty]$ with $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}>0$, and let $w=\prod_{j=1}^m w_j$. Suppose $T$ is well-defined on all $m$-tuples $f_j\in L^{p_j}_{w_j}(\mathbb{R}^d)$ with $\vec{w}\in A_{\vec{p}}$, and suppose there is an increasing function $\phi:[0,\infty)\to[0,\infty)$ such that

$$\|T(f_1,\ldots,f_m)\|_{L^p_w(\mathbb{R}^d)}\leq\phi([\vec{w}]_{\vec{p}})\prod_{j=1}^m\|f_j\|_{L^{p_j}_{w_j}(\mathbb{R}^d)}.$$

Let $X_1,\ldots,X_m$ be Banach function spaces over $\mathbb{R}^d$ for which

$$M_{\vec{m},1}:X_1^{\frac{1}{m}}\times\cdots\times X_m^{\frac{1}{m}}\times(X^{\frac{1}{m}})'\to L^1(\mathbb{R}^d)$$

is bounded, where $X$ is the associated product space. Multilinear extrapolation conjecture. Then $T(f_1,\ldots,f_m)$ is well-defined for every $f_j\in X_j$, and there is an increasing function $\psi:[0,\infty)\to[0,\infty)$ such that

$$\|T(f_1,\ldots,f_m)\|_X\leq\psi\left(\|M_{\vec{m},1}\|_{X_1^{\frac{1}{m}}\times\cdots\times X_m^{\frac{1}{m}}\times(X^{\frac{1}{m}})'\to L^1(\mathbb{R}^d)}\right)\prod_{j=1}^m\|f_j\|_{X_j}.$$

This proposes extrapolation of multilinear weighted bounds to general Banach function spaces, including product spaces that may be quasi-Banach when $p<1$. The supplied text gives no resolution.