The identity-generation conjecture for mutations of perm algebras

Let $A$ be a mutation of a $\mathfrak{perm}$ algebra over a field of characteristic zero, and let $f(a,b,c)$, $\mathcal{WA}(a,b,c)$, $\overline{H}(a,b,c,d)$, and $\overline{I}(a,b,c,d)$ denote the four stated identities. The two additional degree-four identities are

$$\langle \langle \langle a,b \rangle, c \rangle, d \rangle +\langle \langle \langle c,d \rangle, a \rangle, b \rangle=\langle \langle \langle a,d \rangle, c \rangle, b \rangle +\langle \langle \langle c,b \rangle, a \rangle, d \rangle,$$

and

$$\langle \langle a,b \rangle, \langle d,c \rangle \rangle +\langle \langle c,\langle b,a \rangle \rangle, d \rangle +\langle \langle \langle b,a \rangle, c \rangle, d \rangle=\langle \langle \langle a,b \rangle, d \rangle, c \rangle +\langle \langle \langle b,c \rangle, a \rangle, d \rangle +\langle \langle \langle c,a \rangle, b \rangle, d \rangle.$$

Identity-generation conjecture. Every multilinear identity in the mutation of a $\mathfrak{perm}$ algebra over a field of characteristic zero is a consequence of

$$f(a,b,c)=0,\quad \mathcal{WA}(a,b,c)=0,\quad \overline{H}(a,b,c,d)=0,\quad \overline{I}(a,b,c,d)=0,$$

and the two additional degree-four identities displayed above. The claim concerns the identity theory of mutations and is motivated by computations showing that these six identities generate all identities through degree seven; whether they generate every multilinear identity in all degrees remains open.