Sparse form domination implies worse sparse form domination

Let $\vec{r}\in(0,\infty)^m$, $q_0\in(0,\infty)$, and $s\in(q_0,\infty]$. Let $T$ be an operator defined on $m$-tuples of functions. Suppose that, for any $f_1,\ldots,f_m\in L^\infty_c(\mathbf{R}^d)$ and some function $g$, there exists a sparse collection $\mathcal{S}$ such that

$$\|T(\vec{f})\cdot g\|_{L^{q_0}(\mathbf{R}^d)}\lesssim \left(\sum_{Q\in\mathcal{S}}\left(\prod_{j=1}^m\langle f_j\rangle_{r_j,Q}\right)^{q_0}\langle g\rangle^{q_0}_{\frac{1}{\frac{1}{q_0}-\frac{1}{s}},Q}|Q|\right)^{\frac{1}{q_0}}.$$

Sparse form domination conjecture. The same estimate should also hold after replacing $q_0$ by any $q\in(0,q_0]$.

This conjecture asks whether sparse domination in form at exponent $q_0$ automatically yields the corresponding, quantitatively weaker, sparse form estimates at all smaller exponents. The text notes that such replacement would give qualitatively the same weighted bounds, but with a worse exponent; whether the needed flexibility follows automatically is unknown.