Multiplicity-one conjecture for truncated wedges of reduced monomial hypersurfaces

Let $k$ be a field and let

$$X=\operatorname{Spec} k[x_{1},\ldots,x_{n}]/(x_{1}\cdots x_{r})$$

be a reduced monomial hypersurface. For a nonnegative integer $m$, let $\mathcal{W}_{m}(X)$ denote its truncated wedge scheme. An irreducible component means an irreducible component of this scheme. Multiplicity-one conjecture. The multiplicity of $\mathcal{W}_{m}(X)$ along any irreducible component is one. This is motivated by calculations suggesting that truncated wedge schemes have multiplicity one along every component, in contrast with the multinomial multiplicities that can occur for jet schemes. The supplied text gives no proof or resolution.