Kostant's filtration conjecture for the odd Harish-Chandra projection

Let $\mathfrak{g}=\mathfrak{n}_+\oplus\mathfrak{h}\oplus\mathfrak{n}_-$ be a Lie algebra with triangular decomposition, let $\operatorname{Cl}(\mathfrak{g})$ be its Clifford algebra, and let $\theta:\mathfrak{g}\to\operatorname{Cl}(\mathfrak{g})$ be the canonical injection. Let $P\subset(\bigwedge\mathfrak{g})^{\mathfrak{g}}$ be the space of primitive elements, let $q$ be the map from $\bigwedge\mathfrak{g}$ to $\operatorname{Cl}(\mathfrak{g})$, and set $\mathcal{P}=q(P)$. Define

$$\mathcal{P}^{(k)}=\{v\in\mathcal{P}:\operatorname{deg}(v)\leq k\}.$$

Let ${\rm hc}_{\rm odd}:\operatorname{Cl}(\mathfrak{g})\to\operatorname{Cl}(\mathfrak{h})$ be the odd Harish-Chandra projection. Let $\check{\mathfrak{g}}$ be the Lie algebra associated with the dual root system, let $(\check e,\check h,\check f)$ be its principal ${\rm sl}_2$-triple, and identify $\mathfrak{h}^*$ with $\check{\mathfrak{h}}$. Define

$$\mathcal{F}^{(m)}\check{\mathfrak{h}}=\{x\in\check{\mathfrak{h}}:\operatorname{ad}_{\check e}^{m+1}x=0\}.$$

Transport this filtration to $\mathfrak{h}$ via the canonical identification. Kostant's filtration conjecture. For every $k\in\mathbb{N}$,

$${\rm hc}_{\rm odd}(\mathcal{P}^{(2k+1)})=\theta\big(\mathcal{F}^{(k)}\mathfrak{h}\big).$$

The conjecture refines the known equality ${\rm hc}_{\rm odd}(\mathcal{P})=\theta(\mathfrak{h})$ by identifying the degree filtration on primitive invariant exterior elements with the principal-${\rm sl}_2$ filtration on the Cartan subalgebra. It is attributed to Kostant and is presented as an open question in the source.