The authors' bump conjecture for two-weight bilinear forms

Let $(w,\sigma)$ be a pair of weights, set $\lambda_Q=|Q|$ and $p=q$, and let $\mathcal N$ be the best constant in the estimate denoted by (eq:norm) in the source. Let $A$ and $B$ be Young functions satisfying

$$\int_{1/2}^{\infty}\frac{A(t)}{t^p}\frac{dt}{t}<\infty, \qquad \int_{1/2}^{\infty}\frac{B(t)}{t^{p'}}\frac{dt}{t}<\infty.$$

For the Luxembourg norm, define

$$\langle f\rangle_{A,Q}:=\inf\\{\lambda>0:\langle A(f/\lambda)\rangle_Q\le 1\\}.$$

Using the source's weights $u,v$ and parameters $q_0,p_0,r,r'$, define

$$[u,v]_{A,q_0,p,r}=\sup_Q\langle v\rangle_Q^{\frac1p-\frac1{q_0}}\langle u\rangle_Q^{(\frac1p-\frac1{q_0})(r-1)}\frac{\langle u\rangle_Q^{\frac1p}}{\langle u^{\frac1p}\rangle_{A,Q}},$$ $$[v,u]_{B,p_0',p',r'}=\sup_Q\langle u\rangle_Q^{\frac1{p'}-\frac1{p_0'}}\langle v\rangle_Q^{(\frac1{p'}-\frac1{p_0'})(r'-1)}\frac{\langle v\rangle_Q^{\frac1{p'}}}{\langle v^{\frac1{p'}}\rangle_{B,Q}}.$$

Bump conjecture. There is a constant $C>0$, independent of $w$ and $\sigma$, such that

$$\mathcal N\le C\left([u,v]_{A,q_0,p,r}+[v,u]_{B,p_0',p',r'}\right).$$

The paper proposes this estimate as a conjecture and states that it implies the one-supremum conjecture with $A_\infty$ replaced by $A_\infty^{\mathrm{exp}}$ and the separated bump conjecture; no resolution is given.