= Solution
The <conjugate partition> is $\lambda'_j=\#\{i:\lambda_i\geq j\}$, obtained by transposing the <Young diagram>. Transposition sends a <standard Young tableau> $T$ to a standard tableau $T'$ and negates every <Content of a Young-diagram cell>.
Use the <Young orthogonal form>. For an admissible pair $R=s_iT$ and axial distance $d=c_T(i+1)-c_T(i)$, the action on $(w_T,w_R)$ is
$$
A_d=\begin{pmatrix}d^{-1}&\sqrt{1-d^{-2}}\\\sqrt{1-d^{-2}}&-d^{-1}\end{pmatrix}.
$$
In the transposed pair it is $A_{-d}$, whereas in the <tensor product of group representations> with the <sign representation> it is $-A_d$. Their diagonal entries agree, and their off-diagonal entries differ by a sign. Fix a reference tableau and let $\epsilon_T$ be the sign of the unique label permutation from it to $T$. For every admissible swap $\epsilon_R=-\epsilon_T$, so
$$
w_T\otimes1\longmapsto\epsilon_Tw_{T'}
$$
intertwines these two actions. In a nonadmissible pair, transposition exchanges row and column and changes the scalar from $+1$ to $-1$ or conversely, also agreeing with the sign twist. Since adjacent <transpositions> generate $S_n$, this is an isomorphism:
$$
\boxed{V^{\lambda'}\cong V^\lambda\otimes\operatorname{sgn}.}
$$
The parity choice is globally well defined because the label permutation is unique; no arbitrary edge-by-edge phase choices are required.
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