Equality of critical exponents for metric measure spaces with the same heat-kernel exponents

Let $(M_1,d_1,{\Greekmath 0116}_1)$ and $(M_2,d_2,{\Greekmath 0116}_2)$ be metric measure spaces admitting heat kernels satisfying the heat kernel estimate (hk), with the same exponents ${\Greekmath 010B}$ and ${\Greekmath 010C}^*$. The quantities ${\Greekmath 011B}_p^*$ and ${\Greekmath 011B}_p^{\#}$ are the critical exponents defined for these spaces. Equality conjecture. Is it true that

$${\Greekmath 011B}_p^*(M_1,d_1,{\Greekmath 0116}_1)={\Greekmath 011B}_p^*(M_2,d_2,{\Greekmath 0116}_2),$$ $${\Greekmath 011B}_p^{\#}(M_1,d_1,{\Greekmath 0116}_1)={\Greekmath 011B}_p^{\#}(M_2,d_2,{\Greekmath 0116}_2),$$

and

$${\Greekmath 011B}_p^*(M_1,d_1,{\Greekmath 0116}_1)={\Greekmath 011B}_p^{\#}(M_1,d_1,{\Greekmath 0116}_1)?$$

The question asks whether these critical exponents depend only on the common heat-kernel exponents and whether the two definitions of the critical exponent coincide. The source presents this as an open problem; no resolution is given.