Conjecture on positive definiteness of the invariant-theoretic matrix Xi

For positive integers $n$ and $k$, let $\operatorname{STab}((k^n))$ be the set of standard tableaux of rectangular shape $(k^n)$. For each $T\in\operatorname{STab}((k^n))$, let $D_T(X)=\operatorname{wrdet}_{k}(g(T)^{-1}\cdot X)$, and define the $f^{(k^n)}\times f^{(k^n)}$ matrix

$$\Xi_{n,k}=\bigl(\varphi_{n,k}(g(T)^{-1}g(S))\bigr)_{S,T\in\operatorname{STab}((k^n))}.$$

Here $\varphi_{n,k}$ is the function introduced in the preceding construction, and $M_{n,k}^{T_{kn},\operatorname{det}}=\mathbb{C}[\mathfrak{S}_{kn}]\cdot\operatorname{wrdet}_{k}$. Positive-definiteness conjecture. The matrix $\Xi_{n,k}$ is positive definite; in particular, $\operatorname{det}\Xi_{n,k}>0$. Equivalently, $\{D_T(X)\}_{T\in\operatorname{STab}((k^n))}$ is another basis of $M_{n,k}^{T_{kn},\operatorname{det}}$. The preceding discussion establishes only that $\Xi_{n,k}$ is symmetric and positive semidefinite, so the asserted strict positivity and basis property remain conjectural.