Conjecture on the minimal ribbon with prescribed row partition

Let $N$ be fixed, let $\lambda\vdash N$, and consider the subposet of $\mathcal{P}_{N}$ consisting of ribbons $R$ whose row partition is $\operatorname{rows}(R)=\lambda$. For a ribbon $R$, let $[R]$ denote its equivalence class, and write $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_{\ell(\lambda)})$. Conjecture on minimal ribbons with prescribed rows. This subposet has a unique minimal element, namely $[R]$, where

$$R=\langle\lambda_1,\lambda_3,\lambda_5,\ldots,\lambda_{\ell(\lambda)},\ldots,\lambda_6,\lambda_4,\lambda_2\rangle.$$

This conjecture subsumes the preceding proposed description of minimal equitable ribbons. The source presents it as a stronger open statement about the structure of the ribbon subposet of $\mathcal{P}_{N}$; the notation $[R]$ denotes the relevant equivalence class.