Fourier multiplier characterization on the product Hardy space

Let $\lambda$ be a locally integrable function on $\mathbb{R}^d$. Let $\mathcal{F}(\lambda)$ denote its Fourier transform, and let $H^1(\mathbb{R}_+^2\times\mathbb{R}_+^2)$ and $L^1$ be the spaces in the claimed multiplier mapping. For every bounded open set $\Omega$, write $R=I\times J$ for dyadic sub-rectangles of $\Omega$, and set

$$A=\frac{2\pi}{|I|},\qquad B=\frac{2\pi}{|J|}.$$

Fourier multiplier conjecture. The function $\mathcal{F}(\lambda)$ is a Fourier multiplier from $H^1(\mathbb{R}_+^2\times\mathbb{R}_+^2)$ into $L^1$ if and only if there exists a constant $C>0$ such that, for every bounded open set $\Omega$, the displayed square-root expression in the source is strictly less than $C$, with the first sum taken over $k,l\in\mathbb{Z}$ and the second over $k,l\in\mathbb{Z}\setminus\{0\}$, using the integrals involving $\widehat{\lambda}(Ak,Bl)$ and $\mathcal{F}(\Psi)(y_1,y_2)$ exactly as stated.

This is proposed as a characterization of Fourier multipliers on the product Hardy space. The supplied text gives no resolution or supporting status evidence beyond introducing it as a conjecture, so its status remains open.