The skewed projection inequality for weak antichains

Let $A\subseteq[0,1]^n$ be a weak antichain, meaning that no two distinct points $x,y\in A$ satisfy $x_i<y_i$ for every coordinate $i$. For each $i\in[n]$, define

$$\underline{A}_i=A\cap\{(x_1,\ldots,x_n)\in[0,1]^n:x_i=\min\{x_1,\ldots,x_n\}\}.$$

Define the skewed projection $\Delta_i:\underline{A}_i\to[0,1]^{n-1}$ by deleting the $i$th coordinate after subtracting $x_i$ from every remaining coordinate:

$$\Delta_i(x_1,\ldots,x_n)=(x_1-x_i,\ldots,x_{i-1}-x_i,x_{i+1}-x_i,\ldots,x_n-x_i).$$

Skewed projection inequality. If $A$ is a weak antichain, then

$$\mathcal H^{n-1}(A)\leq\sum_{i=1}^{n}\mathcal H^{n-1}\bigl(\Delta_i(\underline{A}_i)\bigr).$$

This proposed extension of projection inequalities replaces orthogonal projections by skewed projections. The source presents it as a belief rather than an established result, and the injectivity of these maps on weak antichains motivates the formulation.