Completeness conjecture for generalized controlled-not identities on Boolean isomorphisms

Let $\mathsf{FinSet}_2$ denote the finite-set prop on the two-element set, and let $\{X,x\}$ denote the generalized controlled-not gate controlled by wires indexed by $X$ and operating on $x$. Let $\operatorname{Iso}(\mathsf{FinSet}_2)$ denote the prop generated by all such gates modulo the following identities:

$$\{X,x\}\{X,x\}=1.$$

If $x\notin Y$ and $y\notin X$, then

$$\{X,x\}\{Y,y\}=\{Y,y\}\{X,x\}.$$

If $x\notin Y$, then

$$\{X,x\}\{\{x\}\sqcup Y,y\}=\{X\cup Y,y\}\{\{x\}\sqcup Y,y\}\{X,x\}.$$

If $x\notin Y$, then

$$\{\{x\}\sqcup Y,y\}\{X,x\}=\{X,x\}\{\{x\}\sqcup Y,y\}\{X\cup Y,y\}.$$

If $x\in Y$ and $y\in X$, then

$$\{X,x\}\{Y,y\}\{X,x\}=\{Y,y\}\{X,x\}\{Y,y\}.$$

Completeness conjecture for generalized controlled-not identities. These identities are complete for $\operatorname{Iso}(\mathsf{FinSet}_2)$: every equality of Boolean isomorphisms generated by generalized controlled-not gates follows from them.

Generalized controlled-not gates generate Boolean isomorphisms, and the conjecture proposes a complete presentation by local identities extending commutation and braid-type relations. The supplied text gives no resolution status, so the conjecture remains open.