The equality characterization conjecture for ribbon Schur Q-functions

Let $\rho_ u$ denote the ribbon Schur $Q$-function indexed by a ribbon $u$. For ribbons $u$, write $u^t$ for the transpose and $u^0$ for the reverse-complement operation, and let $\bullet$ and $\circ$ denote the ribbon concatenation operations. The notation in the conjecture consists of ribbons $alpha_i,\beta_i,gamma_i,delta_i,varepsilon_i,eta_i$ and nonnegative integers $j,k,ell$.

Equality characterization conjecture. For ribbons $\alpha,\beta$, one has $\rho_\alpha=\rho_\beta$ if and only if there exist $j,k,ell$ such that

$$\alpha=\alpha_1bullet\cdots\bulletalpha_j\bullet(\gamma_1\circ\cdots\circgamma_k)\bulletvarepsilon_1\bullet\cdots\bulletvarepsilon_\ell$$

and

$$\beta=\beta_1\bullet\cdots\bulletbeta_j\bullet(\delta_1\circ\cdots\circdelta_k)\bulleteta_1\bullet\cdots\bulleteta_\ell,$$

where

$$\alpha_i,\beta_i\in\{2,11\}\qquad 1\leq i\leq j,$$ $$\delta_i\in\{\gamma_i,\gamma_i^\circ\}\qquad 1\leq i\leq k,$$

and

$$\eta_i\in\{\varepsilon_i,\varepsilon_i^t,\varepsilon_i^\circ,(\varepsilon_i^t)^\circ=(\varepsilon_i^\circ)^t\}\qquad 1\leq i\leq\ell.$$

The conjecture would characterize all equalities between ribbon Schur $Q$-functions by the stated local transformations; one direction is proved in the paper, and the claim had been confirmed computationally for ribbons with up to 13 cells.