Relatively spectral homomorphisms and K-injectivity

Let $\text{A}$ and $\text{B}$ be unital Banach algebras and $\varphi \text{\hspace{0.17em}⁣}:\text{A}\to \text{B}$ be a unital continuous homomorphism. We prove that if $\varphi$ is relatively spectral (i.e., there is a dense subalgebra $X$ of $\text{A}$ such that ${\text{sp}}_{\text{B}}(\varphi (a))={\text{sp}}_{\text{A}}(a)$ for every $a\in X$) and has dense range, then $\varphi$ induces monomorphisms from $K_i(\text{A})$ to $K_i(\text{B})$, $i=0,1$.