Zero-dimensional intersection conjecture for degree-\d hypersurfaces

Let $\kappa$ be an algebraically closed field, and let $S_d(m,\kappa)$ be the degree-$d$ graded component of $\kappa[x_0,\dots,x_m]$. For $m\leq r\leq \binom{d+m}{m}$, define

$$u_r(d,m):=\max\{|V(W)|:W\subseteq S_d(m,\kappa),\ \dim(W)=r,\ \dim(V(W))=0\}.$$

Let

$$\Omega'(d,m):=\Big\{(\alpha_1,\alpha_2,\dots,\alpha_{m+1})\in\mathbb{N}_0^{m+1}:\sum_{i=1}^{m+1}\alpha_i=d,\ d\notin\{\alpha_2,\dots,\alpha_m\}\Big\}.$$

Order $\Omega'(d,m)$ lexicographically, let $\omega'_r(d,m)$ be its $r$th largest element, and, if $\omega'_r(d,m)=(\alpha_1,\dots,\alpha_{m+1})$, define

$$H'_r(d,m):=\alpha_1d^{m-1}+\dots+\alpha_m.$$

Zero-dimensional intersection conjecture. Given $d,m\geq 1$ and $m\leq r\leq \binom{d+m}{m}$, we have

$$u_r(d,m)=H'_{r-(m-1)}(d,m).$$

This conjecture proposes an exact formula for the largest possible cardinality of a zero-dimensional common vanishing set of an $r$-dimensional space of degree-$d$ forms. The supplied text does not establish the conjecture or give evidence resolving it; its status is therefore open.