Precise nonvanishing range conjecture for Veronese syzygies

Fix integers $n\geq0$, $b\geq0$, and $q\in[0,n]$. Define

$$K_{p,q}(n,b;d)=K_{p,q}\bigl(\mathbf{P}^n,\mathcal{O}_{\mathbf{P}^n}(b);\mathcal{O}_{\mathbf{P}^n(d)}\bigr).$$

For $d\geq b+q+1$, the known nonvanishing theorem gives $K_{p,q}(n,b;d)\neq0$ throughout the range

$$\binom{q+d}{q}-\binom{d-b-1}{q}-q\leq p\leq \binom{n+d}{n}-\binom{d+n-q}{n-q}+\binom{n+b}{q+b}-q-1.$$

Precise nonvanishing range conjecture. In the situation of that theorem, one has

$$K_{p,q}(n,b;d)=0$$

when $p$ lies outside the displayed range.

Thus the displayed interval is conjectured to be exactly the nonvanishing range for these Veronese syzygies. The assertion extends the known examples, including the quadratic syzygies of Veronese surfaces, but the supplied source does not report a resolution.