Weak Brumfiel–Hilden conjecture for the A-polynomial ideal

For a knot $K\subset S^3$, let $J\subset \mathbb{C}[m^{\pm1},l^{\pm1}]$ be the kernel of the algebra map determined by the meridian and longitude, and let $A_K(m,l)$ be the $A$-polynomial. Weak Brumfiel–Hilden conjecture. The ideals satisfy

$$\langle J,m-m^{-1}\rangle=\langle A_K(m,l),m-m^{-1}\rangle.$$

This is presented as a weaker version of the Brumfiel–Hilden conjecture that $J=\langle A_K(m,l)\rangle$; the source does not report a resolution, so the weaker equality remains open.