For all $m\gg 1$ set $\varphi_{m}:=\tfrac{1}{m}\log|\mathfrak{a}_{m}|=\tfrac{1}{m}\varphi_{F_{m}}$. This is a $\theta$-psh model function, and [BFJ11, Theorem 8.5] states that $\varphi_{m}\to P_{\theta}(0)$ uniformly on $X$. Unravelling the definitions, we find

$$-\left(\tfrac{1}{m}\mathcal{M}_{m}\right)^{n}\cdot\left(\tfrac{1}{m}F_{m}\right)=\int\varphi_{m}\,(\theta+dd^{c}\varphi_{m})^{n}.$$

By Theorem 3.1 the right-hand side converges to $\int P_{\theta}(0)\left(\theta+dd^{c}P_{\theta}(0)\right)^{n}$, which proves the result. ∎