Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?

The following is a formal capture of that idea:

To the language of $\sf ZF$ (i.e., mono-sorted $\sf FOL(=,\in)$) add primitive partial unary functions $W$ and $j$.

To the axioms of $\sf ZF$, add the following axioms:

Restriction: $\forall \alpha: W_\alpha \lor j_\alpha \to \operatorname {ordinal}(\alpha)$

Injectivity: $W_\alpha \land W_\beta \land \alpha \neq \beta \to W_\alpha \neq W_\beta$

Cumulation: $\forall \operatorname {ordinal} \alpha \exists \lambda: W_\alpha=V_\lambda$

Elementarity: $\forall \operatorname {ordinal} \alpha \, (j_\alpha: W_\alpha \to W_\alpha \land \\ \forall \vec{x} \in W_\alpha [ \phi(\vec x) \leftrightarrow \phi(j_\alpha[\vec x ])] \land \\ \exists x: j_\alpha (x) \neq x) \\\text {where } \phi \text { is purely set theoretic }$

Reflection: $\forall \vec{x} \in W_\alpha \, (\phi \to \phi^{W_\alpha})$

if $\phi$ [in Reflection] is a formula of the language of set theory + $``j_\alpha \!"$, meaning that $W$ doesn't occur in it and every occurence of $j$ must be subscripted with $\alpha$; also $``\alpha \!"$ only appears in $\phi$ as a subscript of $j$.

Where $V_\lambda$ stands for the $\lambda^{th}$ stage of the cumulative hierarchy, defined in the customary manner. $\phi^X$ stands for relativising all quantifiers in $\phi$ to $``\in X\!"$.

Is the above theory consistent relative to some large cardinal property? If so, Which one?