Shao-Yin-Zhao's conjecture for associated varieties at non-admissible integer levels

Let $\fg$ be a simple simply-laced Lie algebra, let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbbm{h}}}=m\geq1$, and let $L_k(\fg)$ be the simple affine vertex algebra. Let $\check{\fg}$ be the Langlands dual Lie algebra, let $\check\BO(m)$ be the nilpotent orbit attached to $m$, and write $\check\BO(m)=\operatorname{Sat}_{\check L}^{\check G}\BO_{\check L}$ from a Bala–Carter Levi $\check\fl$. Let $\fl$ be the corresponding Levi of $\fg$, let $\fp=\fl\oplus\fu$ be a parabolic subalgebra, let $\fz(\fl)$ be the center of $\fl$, and let $\bd$ and $\bd_L$ denote the Barbasch–Vogan–Lusztig–Spaltenstein duality maps. Write $\cN$ for the nilpotent cone. Shao-Yin-Zhao's conjecture. One has

$$X_{L_k(\fg)}=\overline{\operatorname{Ad}G\cdot(\bd_L\BO_{\check L}\times\fz(\fl)\times\fu)}.$$

In particular,

$$X_{L_k(\fg)}\cap\cN=\overline{\bd\check\BO(m)},$$

and $X_{L_k(\fg)}\subseteq\cN$ if and only if $\check\BO(m)$ is distinguished. This is the integer-level predecessor of the paper's main rational-level conjecture; the source recalls it from earlier work and does not report a resolution.