Given theories $T,U$ extending $\textsf{ZFC}$, Steel (p3) defines the consistency strength hierarchy as follows: $T \leqslant_{\text{Con}} U$ iff $\textsf{ZFC} \vdash \operatorname{Con}{U} \to \operatorname{Con}{T}$, where we get strict inequality iff we also have $\textsf{ZFC}\nvdash \operatorname{Con}{T} \to \operatorname{Con}{U}$.

Hamkins (p3) notes that we may give as a sufficient condition for strict inequality that $U \vdash \operatorname{Con}{T}$, also noting (p4) that this is how Simpson (p2) defines his consistency strength relation outright. However these hierarchies are not the same, since (Hamkins p4), for example, the latter order is not dense on extensions of the base theory.

Following from the above, I've been looking for a pair of theories $T,U$ such that we have $T <_{\text{Con}} U$, but $U \nvdash \operatorname{Con}{T}$. More fully, the three conditions I require to be met are:

- $\textsf{ZFC}\vdash \operatorname{Con}{U} \to \operatorname{Con}{T}$,
- $\textsf{ZFC}\nvdash \operatorname{Con}{T} \to \operatorname{Con}{U}$,
- $U \nvdash \operatorname{Con}{T}$.

It seems like theories witnessing the density of Steel's $\leqslant_{\text{Con}}$ over Simpson's would provide the required counter-example, but I'm not aware of these. A related question from math.SE (with no answer) is here.