Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.

This question was proposed (problem A.611) some time ago at KoMaL.
The minimal values for $n=0,1,2,3$ are $3,5,14,324/5$.
It was also discussed at math.se, where the polynomial minimising such sum was found using Gram-Schmidt algorithm.
However the minimum value was not determined.

Any suggestion is welcome.