Extremal $L_p$ interpolation in the mean with intersecting averaging intervals

We find the smallest constant $A=A(n,p,h)$ ($1<h<2$, $1<p<\infty$) such that for any sequence $y_k$, $k\in\mathbb Z$ whose $n$th differences are bounded by one in $l_p$ there is a function $f(x)$ with locally absolutely continuous $(n-1)$th derivative and with $n$th derivative in $L_p(\mathbb R)$ not exceeding $A$ that satisfies the mean interpolation conditions $\frac{1}{h}\,\int _{-h/2}^{h/2}f(k+t)\,dt=y_k$ ($k\in\mathbb Z$). Until now the solution to this problem was known only for non-intersecting averaging intervals ($0\geqslant h\geqslant 1$).