Canonical geometric stochastic heat: global Brownian-loop invariance

Let $(M,g)$ be a smooth compact connected Riemannian manifold without boundary, with scalar curvature $R_g$ for the curvature convention $R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z$, and let $S^1 = \mathbb R/\mathbb Z$. Let $\sigma_1, \dots, \sigma_m$ be smooth vector fields with $$\sum_i \sigma_i \otimes \sigma_i = g^{-1}, \qquad \sum_i \nabla_{\sigma_i}\sigma_i = 0,$$ for instance the tangential projections of the coordinate vectors under a smooth isometric embedding, and let $\xi_1, \dots, \xi_m$ be independent space-time white noises on $\mathbb R_+ \times S^1$. Consider $$\partial_t u = \nabla_{\partial_x u}\partial_x u + \tfrac18 \operatorname{grad}_g R_g(u) + \sqrt2 \sum_i \sigma_i(u)\,\xi_i \tag{1}$$ in the canonical sense of Bruned, Gabriel, Hairer and Zambotti: the solution theory of their Theorem 1.6, whose renormalisation counterterm is fixed by the three conditions there (no additional multiple of $\nabla R_g$ is allowed), applied with metric $g/2$, vector fields $\sqrt2\sigma_i$ and drift $\frac18 \operatorname{grad}_g R_g$. They proved that the canonical solution exists up to a possible explosion time and does not depend on the mollifier or on the choice of the $\sigma_i$.

Let $p_t(x,y)$ be the heat kernel of the diffusion with generator $\frac12\Delta_g$ with respect to Riemannian volume, and $Z = \int_M p_1(x,x)\,d\mathrm{vol}_g(x)$. The Brownian loop probability $\mu_g$ chooses a base point with density $Z^{-1}p_1(x,x)$ and then runs the time-one Brownian bridge from it back to itself; it is supported on $C^\alpha(S^1, M)$ for every $0 < \alpha < \frac12$. Bruned, Gabriel, Hairer and Zambotti conjectured that, for a universal constant $c$, the canonical solution of the equation with drift $c \operatorname{grad}_g R_g$ in place of $\frac18 \operatorname{grad}_g R_g$ exists for all times and has the Brownian loop measure as its unique invariant measure (within each homotopy class, by their Remark 4.6), and that $c = \frac18$; without the factor $\sqrt2$ in front of the noise, the corresponding coefficient, for the Brownian loop measure at half its natural speed, is $\frac1{32}$.

Problem. Is it true that for every such $(M,g)$ and every $0 < \alpha < \frac12$, the maximal canonical solution of (1) does not explode from any deterministic $u_0 \in C^\alpha(S^1, M)$, and $\mu_g$ is invariant for its Markov semigroup, that is, if $u(0) \sim \mu_g$ independently of the driving noises then $u(t) \sim \mu_g$ for every $t \ge 0$?