Weak Segre–Verlinde correspondence and symmetry for four-dimensional affine space

Let $X=\mathbb{C}^4$, let $E=\bigoplus_{i=1}^{N}\mathcal{O}_X\langle y_i\rangle$ and $V=\bigoplus_{i=1}^{r}\mathcal{O}_X\langle v_i\rangle$, and let $\alpha\in K_{\mathsf{T}}(X)$. Weak Segre–Verlinde correspondence and symmetry. For some choice of signs $o(\mathcal{L})$,

$${\mathcal{S}}_X(E,V;(-1)^Nq)={\mathcal{S}}_X(V,E;(-1)^rq),$$

and

$${\mathcal{S}}_{X,0}(E,\alpha;q)-{\mathcal{V}}_{X,0}(E,\alpha;(-1)^Nq)=\sum_{n=1}^\infty \frac{F_n}{(\lambda_1\lambda_2\lambda_3\lambda_4)^{n-2}}\left(\int_X c_3(X)\right)^2q^n,$$

where $F_n\in H_{\mathsf{T}}^{4n-6}(\operatorname{pt})$ depends on $\alpha$ through its rank and Chern classes. The conjecture is motivated by the surface case and was checked computationally in several low-degree and low-rank cases; a general proof is not supplied.