Non-degeneracy conjecture for the intersection pairing on Shtuka isotypic parts

Let $C$ be a smooth projective curve, let $I$ be a finite index set with $|I|=r$, and let $\underline{\e}\in\{\pm1\}^r_0$. For $d\in\ZZ$, the intersection pairing on the degree-$d$ component of the $\GL_n$-Shtuka stack is

$$\langle-,-\rangle_{\underline{\e}}^d:H_c^{nr}(\Sht_{\GL_n,\underline{\e}}^d,\underline{\Qlbar})\otimes H_c^{nr}(\Sht_{\GL_n,\underline{\e}}^d,\underline{\Qlbar})\to\Qlbar.$$

For an irreducible Weil local system $fs_n\in\Loc_{\GL_n}^{\arith}(\Qlbar)$, write $(\prod_{e\in\ZZ}H_c^{nr}(\Sht_{\GL_n,\underline{\e}}^e,\underline{\Qlbar}))_{fs_n}$ for its isotypic part, and write $fs_n^*$ for the dual local system. Non-degeneracy conjecture. For every $d\in\ZZ$, the restricted pairing defines a non-degenerate bilinear form

$$\langle-,-\rangle_{\underline{\e},fs_n}^d:(\prod_{e\in\ZZ}H_c^{nr}(\Sht_{\GL_n,\underline{\e}}^e,\underline{\Qlbar}))_{fs_n}\otimes(\prod_{e\in\ZZ}H_c^{nr}(\Sht_{\GL_n,\underline{\e}}^e,\underline{\Qlbar}))_{fs_n^*}\to\Qlbar.$$

The conjecture predicts perfect duality between the isotypic summands for a local system and its dual; the source gives no evidence of a resolution.