On the Nazarov--Shcheglova Conjecture for Sharp Sobolev Inequalities: The Case (n,p)=(3,2)

For integers $n>k\geq0$ and $1\leq p,q\leq\infty$, let $λ_3(n,k,p,q)$ denote the optimal constant in the one-dimensional Sobolev inequality \[ \|u^{(k)}\|_{L^q(0,1)} \leq λ_3(n,k,p,q) \|u^{(n)}\|_{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1). \] Nazarov and Shcheglova \cite{NazarovShcheglova} conjectu