The higher-dimensional jet ampleness vanishing conjecture for syzygies

Let $X$ be a smooth projective variety of dimension $n$, let $L_d=dA+E$ with $A$ ample and $E$ an arbitrary divisor on $X$, and let $B$ be a line bundle on $X$. For the Koszul cohomology group $K_{p,1}(X,B;L_d)$, the conjecture concerns its vanishing for all sufficiently large $d$. Higher-dimensional jet ampleness conjecture. If $B$ is $p$-jet very ample, meaning that for every effective zero-cycle $z=a_1x_1+\cdots+a_sx_s$ of degree $p+1$, the natural map

$$H^0(X,B)\longrightarrow H^0\left(X,B\otimes\mathcal{O}_X/\mathfrak{m}_1^{a_1}\cdots\mathfrak{m}_s^{a_s}\right)$$

is surjective, then

$$K_{p,1}(X,B;L_d)=0\quad\text{for all }d\gg0.$$

This would give geometric conditions ensuring asymptotic vanishing of the most interesting Koszul cohomology group in higher dimensions. The assertion is presented as a conceivable extension of the curve case, but the supplied text does not establish it.