(0, 1]$ with potential $f$ which satisfies the conclusions of Proposition \[prop:brf\] relative to the conical metric $g_c$ and the parameters $K_0$ and $N_0$. Let ${\mathbf{X}}$, ${\mathbf{Y}}$ be smooth, uniformly bounded families of sections of tensor bundles over $E_{R_0}\times [0, 1]$ with the property that, for any $\epsilon > 0$, there is an $R_1 = R_1(\epsilon) \geq R_0$ such that $$\begin{aligned}
\label{eq:xysys2}
\begin{split}
    \left|{\frac{\partial {\mathbf{X}}}{\partial\tau}} + \Delta{\mathbf{X}}\right| &\leq \epsilon\left(|{\mathbf{X}}| + |{\mathbf{Y}}|\right)\\
    \left|{\frac{\partial {\mathbf{Y}}}{\partial\tau}}\right| &\leq C_0\left(|{\mathbf{X}}| + |\nabla {\mathbf{X}}|\right) + \epsilon|{\mathbf{Y}}|
\end{split}\end{aligned}$$ for some constant $C_0 > 0$ on $E_{R_1}\times [0, 1]$. Then, if ${\mathbf{X}}$ and ${\mathbf{Y}}$ vanish identically on $E_{R_0}\times\{0\}$, they vanish identically on $E_{R_2}\times [0, \tau_0]$ for some $R_2 \geq R_1$ and $\tau_0 \in (0, 1)$.

The two sets of Carleman estimates given in Proposition 4.9 and Propositions 5.7 and 5.9 in [@KotschwarWangConical] are valid on any asymptotically conical shrinking self-similar family of background solutions $g(\tau)$, and the argument in Section 6 of that reference can be applied directly to our setting to combine them to prove the vanishing of ${\mathbf{X}}$ and ${\mathbf{Y}}$. Although the particular components of the system $({\mathbf{X}}, {\mathbf{Y}})$ in [@KotschwarWangConical] – the same as that in [@KotschwarRFBU; @KotschwarFrequency] – differ from those of the system 