Magic permutations for linear forms

The conjecture fails both as stated and under the natural corrected interpretation that the positive forms should take a common value.

Let $d=8$, take the identity permutation, and write

$$J=x_1+x_2+\cdots+x_8.$$

Define three strictly positive linear forms of equal total weight $12$:

$$f_A=J+x_1+x_2+x_7+x_8,$$ $$f_B=J+x_1+x_3+x_4+x_7,$$ $$f_C=J+x_3+x_4+x_5+x_6.$$

Every coefficient belongs to $\{1,2\}$. Their suffix-coefficient vectors in the identity order are

$$v_A=(12,10,8,7,6,5,4,2),$$ $$v_B=(12,10,9,7,5,4,3,1),$$ $$v_C=(12,11,10,8,6,4,2,1).$$

They are pairwise incomparable: $v_A,v_B$ differ in opposite directions at coordinates $3,5$; $v_A,v_C$ at coordinates $2,6$; and $v_B,v_C$ at coordinates $2,7$. Thus the forms satisfy the required reverse-dominance antichain condition.

Literally, the conjectured common-zero subspace cannot contain a positive point because each form is strictly positive throughout the positive orthant. More substantially, replacing common zero by common value does not repair the assertion. Any common-value point must satisfy

$$\begin{aligned} 0&=f_A(x)+f_C(x)-2f_B(x)\\ &=(x_2-x_1)+(x_5-x_3) +(x_6-x_4)+(x_8-x_7). \end{aligned}$$

The right-hand side is strictly positive whenever

$$0<x_1<x_2<\cdots<x_8.$$

Therefore the identity permutation is not realizable.

Nevertheless the common-value subspace contains the pairwise-distinct positive point

$$x=(1,2,3,6,4,5,8,7).$$

Here $J=36$ and

$$f_A(x)=f_B(x)=f_C(x)=54.$$

Thus the antichain criterion fails even for equal-weight strictly positive integer-coefficient forms whose common-value subspace admits a strongly magic labeling.