Relatively spectral homomorphisms and K-injectivity

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

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