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 $z_\mu = \prod_i i^{m_i(\mu)} m_i(\mu)!$ is the standard numerical factor and $p_\mu$ denotes the power-sum symmetric function.

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

A 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. As far as I know, it is still open.

*Some background is given in Luc Lapointe, Luc Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture.*

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