Random 1-factorisation substructure conjecture

Let $\mathcal{U}$ be the Latin square representing a uniformly random 1-factorisation of $K_{2n}$. Let $\ell(\mathcal{U})$ be the length of its longest row cycle, and let $s(\mathcal{U})$ be the size of its largest proper sub-1-factorisation. Random 1-factorisation conjecture. With probability $1-o(1)$,

$$\ell(\mathcal{U})>s(\mathcal{U}).$$

If true, this analogue of the proposed Steiner-quasigroup property would imply average-case polynomial-time canonical labelling for these 1-factorisations. The source gives no evidence of a resolution, so the conjecture remains open.