Finiteness conjecture for Gröbner bases of odd-arity ternary identities

For each $d\in\mathbb{N}$, consider the nonsymmetric operad encoding the ternary identity $w_{2d+1}^{(3)}=0$ and its reduced Gröbner basis.

Ternary Gröbner-basis finiteness conjecture. For each $d\in\mathbb{N}$, the reduced Gröbner basis for the nonsymmetric operad encoding the identity $w_{2d+1}^{(3)}=0$ is finite.

The claim is stated as a further expected result after the ternary multiplication-nilpotence conjecture. Its general validity remains open.