The graduated-order conjecture for the order $H_{n,\lambda}$

Let $K$ be the local field, $R$ its valuation ring, and $\varpi$ a uniformizer. Let $V$ be the underlying $K$-vector space, let $S_\lambda(V)$ be the representation associated with the partition $\lambda$, and write

$$N=\dim_K(S_\lambda(V)).$$

Suppose that the residue characteristic parameter satisfies $\ell=0$. For a matrix $M=(m_{ij})\in\mathbb{Z}^{N\times N}$, define

$$\Lambda_M=\{X\in K^{N\times N}:X_{ij}\in\varpi^{m_{ij}}R\}.$$

Graduated-order conjecture. The order $H_{n,\lambda}$, represented as a matrix order using the standard basis of $S_\lambda(V)$, is a graduated order: there exists a matrix $M=(m_{ij})\in\mathbb{Z}^{N\times N}$ such that

$$m_{ii}=0\quad\text{for }1\leq i\leq N,$$

and

$$m_{ij}\leq m_{ik}+m_{kj}\quad\text{for }1\leq i,j,k\leq N,$$

with

$$H_{n,\lambda}=\Lambda_M.$$

This conjecture gives a concrete matrix description of the order $H_{n,\lambda}$ in mixed characteristic when $\ell=0$ and concerns the structure of the finite convex fixed-point set associated with the representation. Its resolution would identify these orders with graduated orders in the sense attributed to EHNSS22; the source provides no evidence of a resolution.