Arras–Joos conjecture for independent sets in Cartesian products of triangles

Let $G$ be the Cartesian product of $t$ copies of the triangle $K_3$:

$$G=\mathbin{\Box}_{i=1}^t K_3.$$

Write $i(G)$ for the number of independent sets of $G$.

Arras–Joos conjecture. As $t$ tends to infinity,

$$i(G)=(1+o(1))3\cdot 2^{t-1}\cdot 2^{3^{t-1}}\exp\left((3/2)^{3^t-1}\right).$$

Even for this non-bipartite Cartesian product, the asymptotic number of independent sets was not known in the source's discussion; this conjecture gives the proposed asymptotic formula.