Conjecture on the initial ideal associated with an $A_{r-1}$ surface singularity

Let $r\in\mathbf{Z}_{\geq 2}$, let $\mathcal{S}$ be the ring in which the ideal $\mathfrak{a}$ is defined, and let $J_r$ be the ideal generated by the monomials described above: the monomials of $I_r$, the monomials $x_k y_{i_1}\cdots y_{i_k}$ with $1\leq k$, $1\leq i_1\leq\cdots\leq i_k$ and $k+i_k\leq r-1$, and the monomials $x_k y_{i_1}\cdots y_{i_k}z_{j_1}\cdots z_{j_{i_k+k-r+1}}$ with $1\leq k$, $1\leq i_1\leq\cdots\leq i_k$, $2\leq j_1\leq\cdots\leq j_{k+i_k-r+1}$ and $k+i_k\geq r$. The weighted reverse lexicographical ordering is the ordering under consideration. Initial-ideal conjecture. The initial ideal of $\mathfrak{a}$ with respect to the weighted reverse lexicographical ordering is the ideal $J_r$ defined above. This conjecture would describe the leading ideal of the defining ideal of the arc space at an $A_{r-1}$ surface singularity and would support the proposed monomial bases used to determine the dimensions of the vector spaces $\mathcal{A}^0_j$.