The tile-structure conjecture for Prim polynomials

Fix integers $n$ and $p$. A tile is a tuple $(c_{\mathrm{out}},c_{\mathrm{in}},v,\varepsilon)$, where $c_{\mathrm{out}}$ and $c_{\mathrm{in}}$ are carry values, $v=(a_1,\ldots,a_n)$ is an integer vector, and $\varepsilon$ is an integer coefficient. Write $F^i s_{v(T_i)}$ for the $i$-fold Frobenius transform of the Schur character indexed by $v(T_i)$, and let the zero tile be $(0,0,(0,\ldots,0),1)$.

Tile-structure conjecture. For fixed $n$ and $p$, there is a finite list of valid tiles such that $\mathrm{Prim}(d)$ is the sum

$$\sum_{T_0,T_1,\ldots}\prod_i\varepsilon(T_i)F^i s_{v(T_i)},$$

where all but finitely many tiles are zero, $c_{\mathrm{out}}(T_i)=c_{\mathrm{in}}(T_{i+1})$, $c_{\mathrm{in}}(T_0)=\mathrm{END}$, and

$$\sum_i\sum_j p^i a_j(T_i)=2d-2.$$

The conjecture abstracts the observed tile descriptions for even-carry and Nim polynomials. It is supported by computed examples, but the source does not provide a general construction or proof.