Conjectured description of the second half of the connective KO-theory summands

Let $k$ be an index for the spectra $\mathcal{A}_k$, let $\mathcal{E}_{i,k}$ denote the $i$th edge of $\mathcal{\Sigma}^{-2^{k+1}}\mathcal{A}_k$, and let $M_i$, $y_1$, and $z_j$ denote the summands and classes used in the decomposition. For exponents $\varepsilon_j\in\{0,1\}$, set

$$S=\sum \varepsilon_j.$$

The exponent $t$ is defined by

$$t=2^{k-2}-2-2^{i-3}-\sum 2^{j-1}\varepsilon_j,$$

and set

$$D=2^i+4t+8+\sum 2^{j+2}\varepsilon_j.$$

Conjectured second-half description. The $\nu$ summands in the second half of $\mathcal{A}_k$ are $y_1^{2^{k-2}-2}M_{k+1}$ and all

$$y_1^t z_3^{\varepsilon_3}\cdots z_{k-2}^{\varepsilon_{k-2}}z_{k-1}M_i$$

with $4\le i\le k$, $\varepsilon_j\in\{0,1\}$, $\varepsilon_j=0$ if $j<i-1$, and $t=2^{k-2}-2-2^{i-3}-\sum 2^{j-1}\varepsilon_j$. Their corresponding edges are

$$y_1^{2^{k-2}-2}M_{k+1}\leftrightarrow \Sigma^{2^k}\mathcal{E}_{2,k}$$

and

$$y_1^t z_3^{\varepsilon_3}\cdots z_{k-2}^{\varepsilon_{k-2}}z_{k-1}M_i\leftrightarrow \Sigma^D\mathcal{E}_{S+3,i+S}.$$

These constitute exactly the second half of $\Sigma^{-2^{k+1}}\mathcal{A}_k$.