  We show that for a strongly local, regular symmetric Dirichlet form over a
complete, locally compact geodesic metric space, full off-diagonal heat kernel
estimates with walk dimension strictly larger than two (\emph{sub-Gaussian}
estimates) imply the singularity of the energy measures with respect to the
symmetric measure, verifying a conjecture by M.\ T.\ Barlow in [Contemp.\
Math., vol.\ 338, 2003, pp.\ 11--40]. We also prove that in the contrary case
of walk dimension two, i.e., where full off-diagonal \emph{Gaussian} estimates
of the heat kernel hold, the symmetric measure and the energy measures are
mutually absolutely continuous in the sense that a Borel subset of the state
space has measure zero for the symmetric measure if and only if it has measure
zero for the energy measures of all functions in the domain of the Dirichlet
form.
