Existence of Ulrich bundles of rank 8k on the degree-2 Veronese fivefold for odd k<15

Determine whether the degree-2 Veronese fivefold X^5_2 supports Ulrich bundles of rank 8k for each odd integer k with 1 ≤ k ≤ 13. This question arises because, for X^5_2, any Ulrich bundle must have rank divisible by 8, and a construction via the irreducible SL_6-representation of highest weight varpi_2+varpi_3 yields an example of rank 16; it remains to ascertain existence for the other odd multiples of 8 below 120.

Background

The paper classifies Ulrich ranks for Veronese threefolds and discusses the difficulty of determining Ulrich ranks for higher-dimensional Veronese varieties. A general arithmetic constraint shows that on X^5_2 any Ulrich bundle must have rank divisible by 8.

The authors give a concrete example on X^5_2 by constructing an SL_6-equivariant bundle associated to the irreducible representation of highest weight varpi_2+varpi_3; by Borel–Weil, twisting this bundle by O(1) yields an Ulrich bundle of rank 16. However, beyond this example, the existence of Ulrich bundles in other admissible ranks remains unresolved for certain smaller odd multiples of 8.