Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.

We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle p_\mu,p_\nu\rangle_\alpha = \delta_{\mu,\nu} \alpha^{\ell(\mu)} z_\mu$, where $z_\mu$ is the standard numerical factor $z_\mu = \prod_i i^{m_i(\mu)} m_i(\mu)!$ and $p_\mu$ denotes power-sum symmetric function.

For any partition $\lambda$, let us denote by $J_\lambda$ the Jack symmetric function . This function is uniquely determined (up to normalization constant - here I use the same normalization as used in Macdonald's book ) by a triangular expansion with respect to the monomial symmetric functions and by being orthogonal with respect to $\langle \cdot,\cdot\rangle_\alpha$.

Famous conjecture of Stanley states that $f_{\mu,\nu}^\lambda(\alpha) := \langle J_\mu J_\nu,J_\lambda \rangle_\alpha$ is a polynomial in $\alpha$ with non-negative integer coefficients and, as far as I know, it is still open.

Here is my question: is it known that for any positive real value of $\alpha$ the corresponding Littlewood-Richardson coefficients are non-negative $f_{\mu,\nu}^\lambda(\alpha) \geq 0$?