Uniform operator-norm conjecture for the linear BGK spectral approximation

Let $N\in\mathbb{N}$ and let $X_N$ be the finite-dimensional polynomial space with orthogonal projection $\Pi_{X_N}$. Let $\Omega$, $\partial_x$, and $\partial_x^*$ be the operators appearing in the inequalities defining $K_N$. Uniform operator-norm conjecture. There exists a constant $K>0$ independent of $N$ such that

$$\sup_{f\in X_N, f\neq 0} \frac{\|\Omega^{-1/2}\Pi_{X_N}\partial_x^*f\|}{\|f\|}\leq K,$$ $$\sup_{f\in X_N, f\neq 0} \frac{\|\Omega^{-1}\partial_x\Pi_{X_N}\partial_x^*f\|}{\|f\|}\leq K,$$ $$\sup_{f\in X_N, f\neq 0} \frac{\|\Omega^{-1}\Pi_{X_N}\partial_x^*\Pi_{X_N}\partial_x^*f\|}{\|f\|}\leq K,$$ $$\sup_{f\in X_N, f\neq 0} \frac{\|\Omega^{-1}\Pi_{X_N}\partial_x^*\partial_xf\|}{\|f\|}\leq K.$$

The conjecture asserts uniform boundedness of the four operator norms as the spectral truncation parameter $N$ varies; the supplied text gives no resolution or further context for whether these bounds are known.