Balabdaoui–Wellner conjecture on uniform Hermite spline interpolation error

Let $k$ be the spline parameter, let $f$ be a $(k-1)$-times differentiable function whose $(k-1)$st derivative has finite total variation, and let $\mathcal{E}_k(f)$ denote the interpolation error in the Hermite interpolation problem via splines of degree $2k-1$, with knots $0=\tau_0<\tau_1<\cdots<\tau_{2k-4}<\tau_{2k-3}=1$. Balabdaoui–Wellner conjecture. The interpolation error is bounded in the supremum norm independently of the locations of the knots; specifically, for $u\in(0,1)$ and $f_u(t)=(t-u)^{k-1}_{+}/(k-1)!$, there exists $c_k>0$ such that

$$\sup_{u\in(0,1)}\sup_{0<\tau_1<\cdots<\tau_{2k-4}<1}\|\mathcal{E}_k(f_u)\|_\infty\leq c_k.$$

Such a bound is needed to establish that the distances between successive knots have stochastic order $n^{-1/(2k+1)}$ in the associated spline problem. The source does not provide evidence that the conjecture has been resolved.