The arbitrary-degree elimination conjecture for the Rees algebra

Let $R$ be the polynomial ring underlying the construction, let $A=R[a,b,x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $f,g,h_1,\ldots,h_n$ be the polynomials defined recursively by the Jacobian construction in the source. Set

$$L=(f,g,h_1,\ldots,h_n)\subset A.$$

Arbitrary-degree elimination conjecture. For arbitrary $n$, $L$ has projective dimension two and specializes to the defining ideal of the Rees algebra $\mathcal R$.

The claim extends the verified degree-$5$ and degree-$6$ cases to arbitrary degree. The source reports computational evidence in those two cases, but gives no general proof or resolution.