Stirling-number refinement of Lech's inequality for monomial ideals

Let $(R,\mathfrak{m})=k[[x_0,\ldots,x_d]]$, and let $I\subseteq R$ be an $\mathfrak{m}$-primary monomial ideal. Let $\stirlingII{d+1}{k}$ denote the Stirling numbers of the second kind. Then

$$\operatorname{e}(I)\leq\sum_{k=0}^{d}(-1)^k\frac{(d+1-k)!}{\binom{d}{k}}\stirlingII{d+1}{d+1-k}\sum_{1\leq i_1<\cdots<i_k\leq d}\ell\left(\frac{R}{I+(x_{i_1},\ldots,x_{i_k})}\right)$$

and equivalently

$$\operatorname{e}(I)=\sum_{k=0}^{d}(-1)^{d-k}\frac{(k+1)!}{\binom{d}{k}}\stirlingII{d+1}{k+1}\sum_{1\leq i_1<\cdots<i_{d-k}\leq d}\ell\left(\frac{R}{I+(x_{i_1},\ldots,x_{i_{d-k}})}\right).$$

Stirling-number Lech conjecture. The first displayed inequality, together with the equivalent reindexing given in the source, holds for every $\mathfrak{m}$-primary monomial ideal $I$. It is known in dimension three, where $d=2$, but remains conjectural in general.