  For any reduced free product $\mathrm{C}^*$-algebra $(A, \varphi) =(A_1,
\varphi_1) \star (A_2, \varphi_2)$, we prove a boundary rigidity result for the
embedding of $A$ into its associated $\mathrm{C}^*$-algebra $\Delta
\mathbf{T}(A, \varphi)$. This provides new examples of rigid embeddings of
exact $\mathrm{C}^*$-algebras into purely infinite simple nuclear
$\mathrm{C}^*$-algebras.
