Critical Sobolev heat-kernel limit and hyperfunction measure classes

Let $\nu_t^{s,m_0}$ be the heat kernels associated with the $W^s$ norm on loops, viewed as measures on $\operatorname{Hyp}(S^1,G)$, where $s>1/2$, $t>0$, and $m_0>0$. Let $\mu_l$ be the measure at level $l$. Hyperfunction heat-kernel limit conjecture. The measures $\nu_t^{s,m_0}$ have a limit as $s\downarrow1/2$, and the measure class of this limit equals the measure class of $d\mu_l$, where $l=1/(2t)$. As $m_0\downarrow0$, this limit converges to $\mu_l$. This refines the critical-loop conjecture by specifying the limiting measure class and its zero-mode limit.