Extended general structure conjecture for the projected drift

Let Assumption hold, with even $\nu$. Assume that $a_{k,l}^z=0$ whenever $k\not\equiv l\pmod 2$, for every $z\in\{+,-,0,1\}$, and suppose that $P_0\phi[2l]=0$ for every $l\in\mathbb{N}$ such that $2l<\nu$. Here $P_0$ is the projection onto the relevant kernel component, $\phi[2l]$ are the recursively defined correction terms, and $\psi[\nu]$ is the $\nu$th recursively defined coefficient.

Extended general structure conjecture. The projected coefficient satisfies

$$P_0 \psi[\nu](x,y) = o(1) + \sum_{\substack{0 \leq \mu \leq \nu \\ \mu \text{ even}}}\left[a_{\nu+1,\mu + 1}^0 + \sum_{n \geq 2} \sum_{\substack{i_1 + \dots + i_n = \nu + n \\ i_j \neq 1 \text{ for } j \notin \{1,n\} \\ r_1 + \dots + r_n = \mu + n \\ r_j \leq i_j \, \text{all} \, j \\ r_1, r_n \text{ even} \\ r_j \text{ odd for } j \notin \{1,n\} }} (-1)^{n-1}\frac{a_{i_1,r_1}^- \left(\prod_{j = 2}^{n-1} a_{i_j,r_j}^1 \right)a_{i_n,r_n}^+}{(a_{1,1}^1)^{n-1}} \right] x^{\mu +1} \psi_x(x).$$

This extends the preceding conjectural formula to the setting with additional coefficient indices and surviving terms; the supplied text gives no resolution status beyond presenting it as a conjecture.