The logarithmic heat-content asymptotic at the critical singularity

Let $M$ be a compact Riemannian manifold with smooth boundary $\partial M$, let $\delta$ denote the distance to the boundary, let $D$ be an operator of Laplace type with Dirichlet boundary conditions, and let $\chi_1$ and $\chi_2$ be smooth functions on $\mathbb{R}_+$ supported in $[0,b]$ and equal to $1$ near $0$, where $\delta$ is smooth on the collar $\partial M\times[0,b]$. The critical logarithmic heat-content asymptotic. If $\alpha_1<2$, $\alpha_2<2$, and $\alpha_1+\alpha_2=1$, then, as $t\downarrow0$,

$$Q(\delta^{-\alpha_1}\chi_1\circ\delta,\delta^{-\alpha_2}\chi_2\circ\delta,D)(t)=2^{-1}\int_{\partial M}dy\,\log t+o(\log t).$$

This identifies the logarithmic leading term at the critical exponent and is motivated by the interval calculation preceding it; the supplied text gives no evidence resolving the general-manifold assertion, so its status is open.