It is trivial to show that Peano artithmetic ($\mathsf{PA}$) supplemented with the $\omega$-rule is complete. Joseph Shoenfield (`On a Restricted $\omega$-Rule', *Bull. Acad. Polon. Sci.* 7 (1959): 405–7) showed that this is true even if we replace the $\omega$-rule with the recursive $\omega$-rule; i.e., we admit

$$\frac{\phi(\bar{0}), \phi(\bar{1}),\ldots}{\forall x \ \phi(x)}$$

only if there exists a recursive function enumerating the proofs of $\phi(\bar{0}), \phi(\bar{1}),\ldots$.

Is it known whether this result can be strengthened? E.g., is $\mathsf{PA}$ $+$ the primitive recursive $\omega$-rule complete? $\mathsf{PA}$ $+$ the Kalmár-elementary $\omega$-rule?