The nonzero-solution conjecture for distance-$d$ polynomials of $H(d,3)$

Let $H(d,3)$ be the Hamming graph on length-$d$ words over an alphabet of size $3$, and let $\Phi_d(H(d,3))$ be the associated system of polynomials. A solution is all-nonzero if every variable in it is non-zero. The nonzero-solution conjecture. For every integer $d\geq 3$, the system

$$f=0\quad\text{for }f\in\Phi_d(H(d,3))$$

has no all-nonzero solution. The conjecture would imply the expected equality $q(H(d,n))=d+1$ for all $d\geq 3$ and $n\geq 3$ via the preceding theorem, while the case $d=2$ is already known from the corollary.