Lassalle's explicit formula conjecture for Jack polynomial shifted symmetric functions

Let $\lambda$ be a partition and let $rgeq 0$. For the Pieri coefficients $c_i^{\alpha}(\lambda)$, define

$$s_r(\lambda)=\sum_{i=1}^{l(\lambda)+1}\left(\lambda_i-\frac{i-1}{\alpha}\right)^r c_i^{\alpha}(\lambda).$$

Let $d_\rho(\lambda)$ and $z_\rho$ denote the quantities associated with a partition $\rho$ used in the shifted-symmetric-function expansion. Lassalle's explicit formula conjecture. For every partition $\lambda$ and every integer $r\geq 0$, one has

$$s_r(\lambda)=\sum_{i=0}^{\lfloor r/2\rfloor}\sum_{j=0}^{r-2i}\sum_{k=0}^{\min(i,j)}\frac{1}{\alpha^i}\left(1-\frac{1}{\alpha}\right)^{r-2i-j}\binom{|\lambda|+i-1}{i-k}\sum_{|\rho|=j}u_{ijk}^{\rho}(r)\frac{d_\rho(\lambda)}{z_\rho},$$

where the coefficients $u_{ijk}^{\rho}(r)$ are positive integers. The conjecture proposes an explicit positive-integer expansion for the shifted symmetric functions $s_r(\lambda)$ arising from Jack polynomial Pieri coefficients; the source does not provide evidence resolving its status.