Quantum K Whitney relations for partial flag varieties

Let $X=\operatorname{Fl}(r_1,\dots,r_k,n)$ be a partial flag variety with tautological bundles

$$0=\mathcal S_0\subset\mathcal S_1\subset\dots\subset\mathcal S_k\subset\mathcal S_{k+1}=\mathbb C^n,$$

where $\mathcal S_j$ has rank $r_j$, and let $q_j$ denote the quantum parameter associated with the degree in the $j$th direction. For a vector bundle $E$, write $\lambda_y(E)=\sum_i[\Lambda^iE]y^i$.

Quantum K Whitney relations. For $j=1,\dots,k$, the following relations hold in $\operatorname{QK}_T(X)$:

$$\lambda_y(\mathcal S_j)\star\lambda_y(\mathcal S_{j+1}/\mathcal S_j)=\lambda_y(\mathcal S_{j+1})-y^{r_{j+1}-r_j}\frac{q_j}{1-q_j}\det(\mathcal S_{j+1}/\mathcal S_j)\star(\lambda_y(\mathcal S_j)-\lambda_y(\mathcal S_{j-1})).$$

These relations are conjectured to give a Whitney-type presentation of the equivariant quantum K-theory ring of a partial flag variety, extending the corresponding relations in quantum cohomology and describing the quantum deformation of the product of lambda classes.