Sheaf property of the structure pre-sheaf on the noncommutative spectrum

Let $A$ be a homotopical ring and let ${\mathrm{Spec}^{\hspace{-0.4mm}\tiny{\mathtt{NC}}}\hspace{-0.5mm}}(A)$ be its noncommutative spectrum equipped with the structure pre-sheaf. Sheaf conjecture. The structure pre-sheaf on ${\mathrm{Spec}^{\hspace{-0.4mm}\tiny{\mathtt{NC}}}\hspace{-0.5mm}}(A)$ is a sheaf. This would establish that the noncommutative spectrum carries a genuine sheaf of homotopical rings, rather than merely a descendable pre-sheaf; the source gives no resolution of the claim.