Continuing stochastic Ricci flow through zero total area

Let $(\Sigma, g)$ be a connected closed orientable smooth Riemannian surface with Euler characteristic $\chi(\Sigma)$. Fix $0 < \gamma < 2$, $\mu > 0$, distinct points $x_i \in \Sigma$ and real charges $\alpha_i < Q$, where $$Q = \frac2\gamma + \frac\gamma2, \qquad \bar\alpha = \sum_i \alpha_i, \qquad 0 < \bar\alpha - Q\chi(\Sigma) < \gamma.$$ Let $m$ be the finite Liouville area-measure law with these vertex insertions, as in Dubédat and Shen (Section 4.2): the push-forward of the field law $\nu_g^\alpha$ by the Gaussian multiplicative chaos map $X \mapsto M_\gamma(X) = {:}e^{\gamma X}{:}_g\,dV_g$. Let $\mathcal X = \mathcal M_+(\Sigma) \setminus \{0\}$, where $\mathcal M_+(\Sigma)$ is the space of finite nonnegative Borel measures on $\Sigma$ with the weak topology, and let $A$ be the $m$-symmetric stochastic Ricci flow of their Theorem 4.3: the diffusion of the closure of $$\mathcal E(F,G) = \gamma^2 \int_{\mathcal X} \sum_{i,j} (\partial_i q)(\partial_j r)\, a(f_i h_j)\, m(da)$$ on smooth cylinder functions $F(a) = q(a(f_0), \dots, a(f_k))$, $f_0 = 1$, and $G(a) = r(a(h_0), \dots, a(h_l))$, with smooth $f_i, h_j$ and compact support in $\mathcal X$ (total mass bounded away from zero), started from quasi-every state. In particular $A$ is killed when it reaches the zero measure. Dubédat and Shen showed that the total area $A_t(\Sigma)$ is a diffusion with generator $$\gamma^2 x\,\partial_{xx} + \bigl(\gamma(\bar\alpha - Q\chi(\Sigma)) - \mu\gamma^2 x\bigr)\partial_x,$$ which behaves at $0$ like a squared Bessel process of dimension $\delta = \frac2\gamma(\bar\alpha - Q\chi(\Sigma))$; the condition on $\bar\alpha$ says $0 < \delta < 2$, the range in which the total area hits zero and can be continued.

Problem. Does there exist a conservative continuous strong Markov process $\widetilde A$ on $\overline{\mathcal X} = \mathcal X \cup \{0\}$ whose part process on $\mathcal X$ agrees with $A$, up to the usual exceptional-set equivalence, and such that for every finite $T$ $$\int_0^T \mathbf 1_{\{\widetilde A_t(\Sigma) = 0\}}\,dt = 0 \quad \text{almost surely?}$$