Let $\mathcal{PR}$ be the set of primitive recursive functions. Let $\mathcal{PR}(f)$ be $\mathcal{PR}$ which we have amplified by adding (a recursive) $f$ the in the set of initial functions. To make this a true extension, assume $f$ outgrows all primitive recursive functions.

Now, $\mathcal{PR}$ is "dense" in the sense that it has the Ritchie-Cobham property, i.e. every function computable in primitive recursive time is primitive recursive and vice versa.

Does $\mathcal{PR}(f)$ have the Ritchie-Cobham property?

To me it seems natural the extension preserves the property and I would guess there is some "direct" argument for this, which I just fail to see.