Finiteness conjecture for Gröbner bases of $k$-ary identities

Let $k\in\mathbb{N}$ and let $d$ satisfy $d\equiv1\pmod{k-1}$. Consider the nonsymmetric operad encoding the $k$-ary identity $w_d^{(k)}=0$ and its reduced Gröbner basis.

$k$-ary Gröbner-basis finiteness conjecture. For all $k\in\mathbb{N}$ and $d\equiv1\pmod{k-1}$, the reduced Gröbner basis for the nonsymmetric operad encoding the identity $w_d^{(k)}=0$ is finite.

This extends the finite-basis expectations stated for binary and ternary identities. Its validity in the stated range remains open.