Sorting-order characterization conjecture for Stanley coefficients

For a partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $E_\lambda^U$ be the set of $\lambda$-Escher tuples, let $\tau_\lambda:E_\lambda^U\to\mathbb{Q}^{\xi(\lambda)}$ be the core representation, and let $E_\lambda^U(\tau)$ be the subset whose core vector has sorting permutation $\tau$. Let $c_\lambda^U$ denote the Stanley coefficient. Sorting-order characterization conjecture. For every partition $\lambda$ there is a set of sorting orders

$$T_\lambda=\{\tau_1,\ldots,\tau_{N(\lambda)}\},$$

independent of $U$, such that

$$c_\lambda^U=\#\bigl(E_\lambda^U(\tau_1)\cup\cdots\cup E_\lambda^U(\tau_N)\bigr).$$

Thus $T_\lambda$ characterizes the Stanley coefficient $c_\lambda$. The claim is the formal version of the paper’s experimentally motivated assertion that Stanley coefficients are determined by sorting orders rather than by the numerical values of core coordinates; the source reports computational evidence but no resolution.