Fiber-type conjecture for Rees algebras of ideals with alternating basic entry sequences

Let $I$ be the ideal considered in the alternating case, with an alternating basic entry sequence as in the preceding construction, and let $\mathcal{R}_R(I)$ denote its Rees algebra. Fiber-type conjecture. In this case, the Rees algebra $\mathcal{R}_R(I)$ is of fiber type. This claim concerns the defining equations of the Rees algebra: equations coming from the symmetric algebra together with equations of the special fiber should suffice. The provided text does not state whether the claim has been proved or remains open.