Symmetric-function generalization of the Eremenko–Gabrielov conjecture

Let $\lambda$ be a partition with $\ell$ parts and fixed parity for each part. Let $G_\lambda(x/x)$ be the specialized symmetric function defined by substituting $2p_{2i-1}$ for $p_{2i-1}$ and $0$ for $p_{2i}$, and let $Q_\mu(x)$ denote Schur's shifted $Q$-function. Symmetric-function generalization of the Eremenko–Gabrielov conjecture. There are integers $c_1,\dots,c_k$ and integer vectors $\gamma_1,\dots,\gamma_k\in\mathbb{Z}^\ell$ such that

$$(-1)^{r(\lambda)}G_\lambda(x/x)=\sum_{i=1}^k c_i Q_{\frac12(\lambda+\gamma_i)}(x).$$

This conjecture is intended as a symmetric-function refinement of the Eremenko–Gabrielov conjecture. The source presents the definition as possibly requiring modification and gives no resolution status.