  We consider strong uniqueness and thus also existence of strong solutions for
the stochastic heat equation with a multiplicative colored noise term. Here,
the noise is white in time and colored in q dimensional space ($q \geq 1$) with
a singular correlation kernel. The noise coefficient is H\"older continuous in
the solution. We discuss improvements of the sufficient conditions obtained in
Mytnik, Perkins and Sturm (2006) that relate the H\"older coefficient with the
singularity of the correlation kernel of the noise. For this we use new ideas
of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the
stochastic heat equation with multiplicative white noise in one dimension. Our
main result on pathwise uniqueness confirms a conjecture that was put forward
in their paper.
