Find a simply connected full-holonomy Spin(7) manifold with odd Euler characteristic

Find a simply connected compact torsion-free Spin(7) manifold with full Spin(7) holonomy and odd Euler characteristic, thereby realizing the nonzero degree-one class \(\mu_7=\eta\) on a full-holonomy target.

Background

The paper exhibits a compact torsion-free barely Spin(7) manifold obtained as a free antiholomorphic quotient of the Fermat sextic. This manifold has Euler characteristic 1305 and realizes the nonzero class $\mu_7=\eta$, but its holonomy is $SU(4)\rtimes\mathbb Z_2$, not full Spin(7) holonomy.

The authors explicitly identify the stronger unresolved goal of finding a simply connected compact example with full Spin(7) holonomy and odd Euler characteristic. Such an example would extend the nonzero geometric class beyond the barely Spin(7) setting represented by the Fermat quotient.