Algebraic independence of logarithms of distinct primes

Let $m\geq 1$ and let $p_1,\ldots,p_m$ be distinct primes. The field $\mathbb Q(z_1,\ldots,z_m)$ of rational functions in $z_1,\ldots,z_m$ is evaluated at $z_j=\ln p_j$ by the homomorphism

$$\psi:\mathbb Q(z_1,\ldots,z_m)\longrightarrow\mathbb Q(\ln p_1,\ldots,\ln p_m).$$

Algebraic-independence conjecture. The numbers $\ln p_1,\ldots,\ln p_m$ are algebraically independent. Equivalently, $\psi$ is an isomorphism.

The conjecture is presented as a consequence of Schanuel's conjecture: unique factorization gives rational independence of the logarithms, and Schanuel's conjecture would then imply the asserted transcendence degree. Its unconditional status is not established in the supplied text.