$(q_1,q_2)$-quasimetrics bi-Lipschitz equivalent to $1$-quasimetrics

We prove that the conditions of $(q_1,1)$- and $(1,q_2)$-quasimertricity of a distance function $\rho$ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to $\rho$. It follows that the Box-quasimetric defined with the use of basis vector fields of class $C^1$ whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of $(q_1,q_2)$-quasimetrics for which there are no Lipschitz equivalent $1$-quasimetrics, which in particular implies another proof of a result by V. Schröder.