Injectivity conjecture for Markov tree fraction labelings

Fix $\sigma\in \mathfrak S_3$ and consider the Markov tree $\mathrm{M}\mathbb T(0,0,0,\sigma)$. For any irreducible fractions $t,s\in[0,1]$, let $n_t$ and $n_s$ be Markov numbers with fraction labelings $t$ and $s$. Injectivity conjecture. If $n_t=n_s$, then $t=s$. The source states that this conjecture is equivalent to Frobenius's uniqueness conjecture and uses it as the classical model for the generalized version.