Lemmermeyer's reflection conjecture for quartic fields and cubic resolvents

Let $L$ be a number field with Galois group $S_4$ or $A_4$ over $\mathbb{Q}$. Let $K_1$ be a quartic subfield and $K_2$ its cubic resolvent. For a number field $K$, write $\mathrm{rk}_2Cl(K)$ for the dimension over $\mathbb{F}_2$ of the $2$-torsion subgroup of its class group. Lemmermeyer's reflection conjecture. In the above setting,

$$\mathrm{rk}_2Cl(K_1)-2\leq\mathrm{rk}_2Cl(K_2)\leq\mathrm{rk}_2Cl(K_1).$$

This sharpens a result of Tsimerman, which gives equality up to an error term depending on the discriminant of $L$. The source does not state whether Lemmermeyer's conjecture has been resolved.