Let $\mathsf{wK4}$ be the logic of weak transitive frames and $\mathsf{GL}$ be the provability logic.
It is well-known that one can define $\mathsf{GL}$ as $\mathsf{K4}$ (the logic of transitive frames) plus Lob's rule: from $\Box A \rightarrow A$ deduce $A$.
But what about $\mathsf{wK4}$ plus Lob's rule? It seems to me that it is not equal to $\mathsf{GL}$. But I cannot come up with a proper explanation.
Any hints or suggestions?
P.S. I am familiar with the Kripke and topological semantics of $\mathsf{wK4}$ and $\mathsf{GL}$.