Is $DSPACE(f) \subseteq DTIME(f)$ always? Is $DSPACE(f) \subseteq DTIME(f)$ always?

For example, if we have a language $A\in DSPACE(log^2(n))$ can we say that $A\in P$ (and  subsequently in NP and coNP) since $DSPACE(log^2(n))\subseteq DSPACE(n^3) \subseteq DTIME(n^3)$?

I know about $ DTIME(f) \subseteq DSPACE(f) \subseteq DTIME(2^{O(f)})$ but $DTIME(2^{O(f)})$ is too big in this case.
We don't know that $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$ is ever true and we suspect that it's never true.

A Turing machine that uses space $f(n)$ can run for $2^{O(f(n))}$ steps without getting stuck in an infinite loop, so it should be able to do much more than a Turing machine that can only run for $f(n)$ steps.  As you've observed, if $\mathrm{DSPACE}[f(n)]\subseteq\mathrm{DTIME}[f(n)]$, then $\mathrm{DSPACE}[f(n)]=\mathrm{DTIME}[f(n)]$ and we have no reason to think that would ever be true for reasonable functions.