Conjectural formula for the multivariate regression-depth constant

Let $d$ and $0\leq k<d$ be constants. For a set of points with $k$ independent and $d-k$ dependent degrees of freedom, let $R(d,k)$ denote a constant such that some $k$-flat has regression depth at least $\lceil n/R(d,k)\rceil$. Conjectured formula for $R(d,k)$.

$$R(d,k)=(k+1)(d-k)+1.$$

The paper proves that such a constant exists and gives an $O(d)$ upper bound for fixed $k$, but says that exact values are unknown for intermediate $k$; the displayed formula is motivated by the known boundary cases and available bounds.