Nonexistence of a Koszul factorization realizing the new degree-33 class

Determine whether any Koszul factorization on the degree-33 Fermat fourfold X_{33}^4 can have a Chern character whose primitive component has a nonzero coefficient of the monomial x_0^{18}x_1^{6}x_2^{12}x_3^{9}x_4^{27}x_5^{21}; specifically, prove the conjectured nonexistence of such a Koszul factorization.

Background

The paper proves the Hodge conjecture for the degree-33 Fermat fourfold by pulling back algebraic classes from a special cubic fourfold. This produces new algebraic classes whose primitive Chern characters include the monomial x_0^{18}x_1^{6}x_2^{12}x_3^{9}x_4^{27}x_5^{21}.

The authors ask whether these classes can instead be realized by an arithmetically Cohen–Macaulay sheaf arising from a Koszul matrix factorization. They conjecture that no Koszul factorization on X_{33}^4 produces a primitive Chern character with a nonzero coefficient of this monomial; the remark reports that prior computational searches had not found one.