Even-row lattice-path model for bounded-height standard Young tableaux

Let $\ell\geq 1$. Let $\{\mathbf{e}_1,\dots,\mathbf{e}_{\ell+1}\}$ be the standard basis of $\mathbb{R}^{\ell+1}$, and let $\mathcal{L}_{2\ell+1}(n)$ be the set of $n$-step lattice paths in $\mathbb{R}_{\geq 0}^{\ell+1}$ from the origin to the axis along $\mathbf{e}_1$, using the $2\ell+1$ step types $\mathbf{e}_1$, $\mathbf{e}_1\pm\mathbf{e}_2$, $\mathbf{e}_1\pm(\mathbf{e}_2-\mathbf{e}_3)$, $\mathbf{e}_1\pm(\mathbf{e}_3-\mathbf{e}_4)$, and so on through $\mathbf{e}_1\pm(\mathbf{e}_{\ell}-\mathbf{e}_{\ell+1})$. Let $\mathcal{L}_{2\ell}(n)$ be the \subset in which the $\mathbf{e}_1$ steps occur only on the hyperplane spanned by $\{\mathbf{e}_1,\dots,\mathbf{e}_\ell\}$. Finally, let $\mathcal{T}_{2\ell}(n)$ be the set of standard Young tableaux with $n$ entries and at most $2\ell$ rows.

Even-row lattice-path conjecture.

$$|\mathcal{T}_{2\ell}(n)|=|\mathcal{L}_{2\ell}(n)|.$$

The conjecture proposes an unexpected relation between the enumeration of standard Young tableaux of even bounded height and constrained lattice paths. The preceding odd-row equality is attributed in the source to work of Grabiner, Magyar, Gessel, and Zeilberger; the even-row analogue is presented as the conjectural step.