Relatively spectral homomorphisms and K-injectivity

Let $\A$ and $\B$ be unital Banach algebras and $\phi\colon\A\to\B$ be a unital continuous homomorphism. We prove that if $\phi$ is relatively spectral (i.e., there is a dense subalgebra $X$ of $\A$ such that $\sp_\B(\phi(a))=\sp_\A(a)$ for every $a\in X$) and has dense range, then $\phi$ induces monomorphisms from $K_i(\A)$ to $K_i(\B)$, $i=0,1$.