Generic-rank conjecture for simultaneous Waring decompositions

Let $n,r,d_{1},\ldots,d_{r}$ be the parameters defining a polynomial vector $f=(f_{1},\ldots,f_{r})$ of forms in $n+1$ variables, where the $i$th component has degree $d_i$. The simultaneous Waring rank is the least number of common linear forms needed to express all components as sums of their respective powers. Write $\lceil x\rceil$ for the least integer greater than or equal to $x$.

Generic simultaneous Waring rank conjecture. The complex simultaneous Waring rank of a general polynomial vector is

$$k=\left\lceil\frac{\displaystyle\sum_{i=1}^{r}\binom{n+d_i}{d_i}}{n+r}\right\rceil.$$

This is the expected dimension-count value and would extend the generic-rank principle known in the ordinary Waring setting. The simultaneous defectivity problem remains open in general, so the conjecture is open; the formula is proved under the paper's nondefectivity and perfect-case assumptions.