Eu's lattice-path enumeration conjecture for standard Young tableaux with at most 2d rows

Let $d$ be a positive integer, and let the admissible unit steps be

$$\{\epsilon_1\}\cup\{\epsilon_1+\epsilon_2,\epsilon_1-\epsilon_2\}\cup\{\epsilon_1-\epsilon_i+\epsilon_{i+1}, \epsilon_1+\epsilon_i-\epsilon_{i+1}:2\le i\le d\},$$

where $\{\epsilon_1,\dots,\epsilon_{d+1}\}$ is the standard basis of $\mathbb{R}^{d+1}$. Consider paths from the origin to $(n,0,\dots,0)$ that stay within the nonnegative octant, with the additional restriction that the steps $\epsilon_1$ are confined to the hyperplane spanned by $\{\epsilon_1,\dots,\epsilon_d\}$. Eu's conjecture. The number of such paths equals the number of $n$-cell standard Young tableaux with at most $2d$ rows. This conjecture refines the established enumeration for standard Young tableaux with at most $2d+1$ rows by imposing the stated hyperplane restriction on the $\epsilon_1$ steps.