Canonical-form conjecture for natural reorderings of finite connected quandles

Let $Q$ be a finite connected quandle with elements $1,2,\ldots,n$ naturally ordered, profile $\{1,\ell_1,\ell_2,\ldots,\ell_k\}$, and $1\leq\ell_1\leq\ell_2\leq\cdots\leq\ell_k$. Let $\nu$ be a natural reordering with respect to $r_n$, as in the last sentence of Theorem. Canonical-form conjecture. In the last sentence of Theorem, one can choose $\nu$ so that $\nu(i)=i$ for every integer $i$ satisfying

$$\left(\sum_{j=1}^{k-1}\ell_j\right)+1\leq i\leq n$$

even when $\ell_{k-1}=\ell_k$. This concerns whether the canonical-form construction can be strengthened when the two largest profile parts are equal. The supplied text gives no evidence that the assertion has been proved or disproved, so it remains open.