I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.

For context, $v(t,r)$ is radial and solves a septic nonlinear wave equation $$-\partial_{tt}v + \Delta v = F(v),$$ on $(t,x)\in[T, T_1]\times \mathbb{R}^3$ where $F(v) = G(v) + g(r,t)$, where $G(0)=0$ and $g(r,t)$ is supported on $\{r=|x|<t+C\}$. We know that at time $T$, $v$ is supported in the ball $\{r=|x|\le T+C\}$ with $C>0$. The authors then claim, on page 34, that by Huygens' principle, it follows $v(r,t)$ is supported in the ball $\{r=|x|<t+C\}$ for each fixed $t\ge T$

However, I believe this simply follows from the finite propagation speed property of the wave equation. Is this correct, or is the Huygens' principle genuinely needed here?