= Solution
Take independent random variables $U_1,\ldots,U_d$, each uniform on $[0,1/2]$, and independent <Rademacher random variables> $\varepsilon_1,\ldots,\varepsilon_{d-1}$. Define
$$
X_i=\varepsilon_iU_i\quad(1\leq i<d),
\qquad
X_d=\left(\prod_{i=1}^{d-1}\varepsilon_i\right)U_d,
$$
and let $\eta$ be the joint law of $(X_1,\ldots,X_d)$.
Each $X_i$ is uniform on $[-1/2,1/2]$. If the $d$th coordinate is omitted, the remaining signs are the independent $\varepsilon_i$. If coordinate $j<d$ is omitted, the remaining signs are
$$
(\varepsilon_i)_{i\ne j},
\qquad
\prod_{i=1}^{d-1}\varepsilon_i,
$$
which are again uniformly distributed over all $2^{d-1}$ sign vectors because this map from $(\varepsilon_1,\ldots,\varepsilon_{d-1})$ is a bijection. The independent magnitudes then show that every $(d-1)$-coordinate marginal is Lebesgue measure.
The full law is not $d$-dimensional Lebesgue measure, because
$$
\prod_{i=1}^d\operatorname{sgn}(X_i)=1
$$
almost surely, whereas this product is equally likely to have either sign under independent Lebesgue coordinates.
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