Strong Fröberg–Iarrobino conjecture for homogeneous linear systems

Let ${\mathcal L}={\mathcal L}_{n,d}(m^s)$ be a homogeneous linear system of degree-$d$ hypersurfaces in ${\mathbb P}^n$ through $s$ general points, each with multiplicity $m$, and let $\operatorname{ldim}({\mathcal L})$ denote its linear expected dimension. Strong Fröberg–Iarrobino conjecture. The system satisfies

$$\dim({\mathcal L})=\operatorname{ldim}({\mathcal L})$$

except perhaps when one of the following conditions holds: $s=n+3$; $s=n+4$; $n=2$ and $s=7$ or $s=8$; $n=3$, $s=9$ and $d\geq 2m$; or $n=4$, $s=14$ and $d=2m$ with $m=2$ or $3$. The conjecture refines the weak lower-bound statement by listing possible exceptional families; the supplied text gives no resolution status.